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= = =< = > ee EE ese in SSS ~ oe = SSS SSS SS RO, Mass We! A AEN A 4k ny a 1 ew Uu | ál 1 7 worl as a are See ref „Teri ilk! $ ik ul vs | Malatins nde oP nn Pars 5 OON LE ENA bj h i Ke ge J hal ve i p Ta ee) LN hs ach ià 1 hd A4 | + m kN Kd 4 ‘ ij | ; ‘ an : on LITERARY KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN -- TE AMSTERDAM -:- Pos ICEEDINGS OF THE £ spec HON OF SCIENCES VOLUME Ix (= 2 PART —} JOHANNES MULLER :—: AMSTERDAM JULY 1907 ; a OE en en d . : YROTELK Krt ie Bh pen End, white her bink coe SARRRRS 172% = ' s % 0 1 6h, Sect Tf (Translated from: Verslagen van de Gewone Vergaderingen der Wis- en Natu Afdeeling van 29 December 1906 tot 26 April 1907. Dl. XV.) r 7 i, % 7 =a b | a 7 CON. TE NTS: Proceedings of the Meeting of December 29 1906 . ....... my > >» > » » January 26 A eo rey at ae ke > >» » > » February 23 » tb ee een > >» » > » March 30 > pe RN oat Wee zone Be > o> > » April 26 » 799 KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM, PROCEEDINGS OF THE MEETING of Saturday December 29, 1906. (Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige Afdeeling van Zaterdag 29 December 1906, Dl. XV). ETE EAN ES: Max Weser: “On the fresh-water fish-fauna of New-Guinea”, p. 462. N. H. Conen: “On Lupeol”. (Communicated by Prof. P. van RomsvrcH), p. 466. N. H. Coven: “On «- and 2-amyrin from bresk”. (Communicated by Prof. P. van RomBurGuy p. 471. F. M. JAEGER: “On substances which possess more than one stable liquid state. and on the phenomena observed in anisotropous liquids”. (Communicated by Prof. A. P. N. FRANCHIMONT), p. 472. F. M. Jarcer: “On irreversible phase-transitions in substances which may exhibit more than one liquid condition”. (Communicated by Prof. A. P. N. Francuimoyr), p. 483. O. Postma: “Some additional remarks on the quantity H and Maxwerr?’s distribution of velocities”. (Communicated by Prof. H. A. Lorentz), p. 492. H. KAMERLINGH Onnes and W. H. Krrsom: “Contributions to the knowledge of the Y-surface of vAN DER Waars. XII. On the gas phase sinking in the liquid phase for binary mixtures”, p. 501. (With one plate). W.H. Kersom: “Contribution to the knowledge of the {-surface of vAN DER Waars. XIII. On the conditions for the sinking and again rising of the gas phase in the liquid phase for binary mixtures”. (Communicated by Prof. H. KAMERLINGH ONNES), p. 508, Erratum, p. 511. nn Nt = a _ Proceedings Royal Acad. Amsterdam. Vol. IX, ( 462 ) Zoology. — “On the fresh-water fishefauna of New Guinea”. By Prof. Max WeBER. (Communicated in the meeting of November 24, 1906). In the year 1877 there appeared a “Quatrième mémoire sur la faune ichthyologique de la Nouvelle-Guinée”, written by P. J. BrLREKER and containing 841 species. These species are exclusively marine and brackish-water fishes and shew clearly, as might be expected, that the littoral fish-fauna of New Guinea belongs to the great Indo-Pacific fauna which extends from the East coast of Africa to the islands of the Western Pacific. The same result is arrived at from the lists published by W. Macrray in 1876 and 1882, which treat of the fishes of the South coast of New Guinea and Torres Straits. But none of these lists accomplished what BLEEKER desired, namely, to give some insight into the nature of the fresh-water fish-fauna of New Guinea. The information which Brreker desired was partly supplied by certain communications, published by W. Macrrar, E. P. Ramsay, J. Dov- GLAS OeiBr, A. Pervera and G. BoureNGer, about fishes caught in the Strickland, Goldie and Paumomu rivers, and in a number of rivulets all situated in the south-eastern part of the island. The number of fishes mentioned amount to about 30, but so long as the fish-fauna of German and Dutch New Guinea remained unknown, it was impossible to give a complete idea of the ichthyological fauna of this big island. This was the more to be regretted inasmuch as fresh-water fishes are of very great assistance in solving zoo-geographical problems. In using them for this purpose we should however keep well in mind the following points. If in regions, at present separated by the sea, identical or closely allied fresh-water forms are found, to which the sea affords an insur- mountable barrier, one may freely draw the conclusion that these regions were formerly either directly or indirectly connected. Among the fresh-water fishes there are however whole categories which cannot be used as factors in such an argument or only with great caution. These are the migratory fishes and those that can live also in brackish water and indeed even in sea-water. The so-called law of E. von Marrens states that from the Poles to the Equator the number of brackish water animals increases. This is also true for fishes and especially for those of the Indo- Australian Archipelago, and in a very remarkable degree for those of the islands east of Borneo and Java. The great Sunda Islands ( 463 ) in consequence of their former connection with the continent of Asia possess a fish-fauna of which the most important elements, both as regards quality and quantity, had no chance of further distribution in an eastern direction. The rivers of the eastern islands of the Archipelago were therefore almost devoid of fishes, and offered a good place of abode for such forms as, though denizens of the sea or of brackish water, possessed sufficient capacity for accommodating themselves to a life in fresh-water. The competition of those Asiatic forms (Cyprinidae, Mastacembelidae, Ophiocephalidae, Labyrinthici ete.), originally better fitted for a fresh-water life, failing, everything was in favour of the immigrants from the sea. The river-fishes of Celebes favour this view, as also does all that we know about the fishes of Ternate, Ambon, Halmahera, etc. We observe the same phenomenon in the fresh waters of Australia. These however contain also indigenous forms, partly very old, partly younger forms; the latter were obviously, at least in part, marine immigrants, which have accommodated themselves so entirely to a fresh-water life as to adopt the characters of fresh-water fishes. The fauna of Australia enjoy at present a general and vivid interest — are there not even people who believe that the cradle of mankind stood there? A remarkable point of interest in the study of its fauna is the question how long Australia has been isolated from other parts of the globe. New Guinea plays a pro- minent rôle in answering this question. It is therefore a welcome fact that the Dutch New Guinea Expe- dition of 1903 under the direction of Prof. A. WicHManNN has brought home; besides other treasures, a large collection of fishes from diffe- rent lakes and many rivers and rivulets, giving us a good insight into the fresh-water fauna of the northern part of the island. It*was of great help to me, while studying this collection, that I was able to make use of the fishes collected in the brackish water at the mouth of the Merauke river, by Dr. Kocu the medical man of the Royal Geographical Society’s Expedition to South New Guinea. The results of this investigation will be published elsewhere, but some more general conclusions may be mentioned here. When we reckon up all the fishes known up to the present date from the lakes, rivers, and rivulets of New Guinea, we find that their number amounts to more than 100 species, but only about 40 of these were found exclusively in fresh-water. A careful examination shews further that the latter species, with a few exceptions, are either known from brackish or sea water at other places, or that their nearest relatives may be found in brac- ale ( 464 ) kish or sea water. New Guinea shews clearly the fact that immi- gration from the sea or from brackish water has played and perhaps still plays a predominant part in the populating of its rivers. Let us now return to the point at issue: namely, that the marine fish-fauna of New Guinea forms part of the great Indo-Pacific fish- fauna and particularly of that of the Indo-Australian Archipelago. Keeping this in mind one might be inclined to draw the conclusion that there is not much to be learned from the fauna of the rivers of New Guinea concerning the history of this island. Such a con- clusion however would be erroneous, for it is clear that the very fishes which are characteristic of the fresh-water of New Guinea belong: 1. to genera which outside New Guinea are known only from Australia (Pseudomugil, Rhombatractus, Melanotaenia, Eumeda) ; 2. or to genera nearly related to exclusively Australian genera. Lambertia for instance is nearly related to Eumeda; Glossolepis to Rhombatractus and the three new species of Apogon are closely allied to Australian ones. Finally the species of Hemipimelodus from New Guinea form a special group, distinct from those of the neighbouring Indian Archipelago. Everything that gives to the fresh-water fish-fauna of New Guinea a character different from that of the Indian Archipelago is at the same time characteristic of Australia. Twelve of its species belonging to the genera Pseudo- mugil, Rhombatractus, Melanotaenia, Glossolepis, belong to the family or subfamily of the Melanotaenidae, only known from Australia. I do not hesitate therefore to maintain that the river-fishes of New Guinea belong to two groups: 1. A fluvio-marine group, which is Indo-Australian or, if one prefers, Indo-Pacific and which may also be met with, for instance, in Ambon or Celebes. To this category belongs also Rhiacichthys (Platy ptera) novae- guineae Blgr. discovered by Pratt in mountain rivers of the Owen Stanley Range four thousand feet high. Boulenger speaks of the disco- very of a fish of the genus Rhiacichthys “so admirably adapted to life in mountain torrents” as highly interesting. He tells us that the closely allied Rhiacichthys asper C. V. is known from Bantam, Celebes and Luzon. This is likely to create the impression that Rhiacichthys novae- guineae does not belong to this category, but is a species whose nearest relative is confined to rivers in regions occupied by the Asiatic fauna. Rhiacichthys asper however, differing but little from Rhiacichthys novae- guineae, was also found by BrrEKER in Sumatra and, what is far more interesting, it occurs, according to Günther, also in Wanderer Bay on the island of Guadaleanar in the Solomon Islands — in “fresh-water”. At all events it is thus found close to the sea. This ( 465 is also true for a specimen which I deseribed from Ambon and still more so for a specimen that I caught near Balangnipa in the lower part of the Tangka, close to its mouth in the gulf of Boni. The water was here already brackish and ran slowly. Rhiacichtys has therefore a very wide distribution, it does not fear brackish water, and its presence in New Guinea loses therewith much of its importance. 2. The second group, the characteristic element, is Australian. This last group requires further explanation as to its origin. In the present state of things, now that New Guinea is separated from Australia by Torres Straits, these offer a barrier impassable to those fishes which I called characteristic. Some species of Rhombatractus and Melanotaenia may it is true, descend to the mouth of the river and be able to endure even slightly brackish water, but none of the 24 recorded species is known from the sea. The barrier can therefore not be bridged by the group of islands in the Torres Straits. They are too poorly supplied with fresh-water and far too strictly coral islands, even when we leave out of consideration the fact that they are separated from each other, from New Guinea and from Australia by broad tracts of sea with a high salt percentage and strong tidal currents. The simultaneous presence of these characteristic forms in New Guinea and in Australia cannot be explained otherwise than by the existence of a more solid and extensive connection in former ages. This connection must have been so far back in the past that, to take an instance, the representatives of the abovenamed Melano- taeniideae had time to separate themselves specifically. And this actually happened; for among the 12 species of Melanotaeniidae already known from New Guinea and among the 12 species described from tropical or sub-tropical Australia not one is common to the two regions, although the differences between some species are very small. On the other hand therefore it cannot have been so very long ago from a geological point of view that this connection between Australia and New Guinea existed. How long a time may have elapsed since that period is at present a matter of hypothesis. But if zoo-geographical and more particularly ichthyological experience may venture an opinion, I should seek the period of this connection not earlier than in the pliocene, and the breaking up of it in the pleistocene. Other zoological observations may perhaps be in favour of this supposition. It will be a long time yet before the last word is spoken on this question. We may express the hope that the new expedition to Dutch Southern New Guinea under the guidance of Dr. H. A. Lorentz, which intends to investigate especially its big rivers, will bring us further light, ( 466 ) Chemistry. — “On Lupeol’'). By Dr. N. H. Conny. (Communi- cated by Prof. P. v. Rompuren). (Communicated in the meeting of November 24, 1906). Notwithstanding the many and beautiful researches of several chemists, the structure of cholesterol, which is important also from a physiological point of view, is far from being known. Therefore, Prof. vaN RomBuran invited me to investigate a substance closely connected with the same, namely lupeol, a phytosterol. For the phytosterols may be included with the cholesterols in one common group “the cholesterollie substances”. The original intention was to study the alstol found by Sack *) in “bresk”*). From the “bresk” investigated by me, alstol, alstonol and #soalstonol could not be iso- lated, although Sack claims to have found them in the same, but I obtained a- and g-amyrin and lupeol. It appeared afterwards that Sack’s alstol is not a chemical individual. Lupeol was first found by Liktmrnik*) in the skins of lupin seeds; afterwards Sack °) met with it in the bark of Roucheria Griffithiana, whilst -vaN RoMmBeren and vAN DER LINDEN ®) demonstrated its presence as a cinnamate in the resin of Palaquium calophyllum. Finally, VAN Rompercu proved that Tscrircn’s’) erystal-albane simply consisted of lupeol cinnamate. The lupeol was prepared from ‘“bresk” by extracting the same first with boiling alcohol. On cooling, a white mass was deposited which, without any further purification was saponified with alcoholic potassium hydroxide. The saponified product was then benzoylated with benzoyl chloride and pyridine and the reaction product treated repeatedly with acetone by heating just to boiling on the waterbath and then filtering off without delay. Finally, a lupeol benzoate was left, which after repeated recrystal- lisation from acetone, consisted of fine, flat needles; m.p. 265°—266°, (corr. 273°—274°). Found C 83.71—83.81 Caleulated for C,,H,,O, 84.07 H 10.41—10.36 10.03 These, like all subsequent combustions, were made with lead chromate. [e]p = + 60°,75 in chloroform. 1) For a more elaborate description see Dissertation N. H. Conen. 1906, Utrecht. 2) Sack. Diss. 1901, Göttingen. 8) Bresk or djetulung is the dried milky juice of some varieties of Dyera. 4) Ztschr. f. physiol. Chem. 15. 415 (1891). 6) Sack 1.c, 6) Ber. 37. 3440 (1904). 1) Arch. der Pharm. 241. 653 (1903). ( 467 ) By saponification of lupeol benzoate with alcoholic potassium hydroxide and recrystallisation from alcohol or acetone, the lupeol was obtained in the form of fine, long needles m.p. 211°, (corr. 215°). Found: © 84.62 84.65 84.40 84.50 Calculated for C,, H,, O 84.85 oo 11.95 11:82 12.02 11.49 lelp = + 27°,2 in chloroform. In the first place it seemed to me of importance to ascertain whether double bonds occur in lupeol. Therefore, a solution of lupeol in carbon disulphide was treated with a solution of bromine in the same solvent. Hydrogen bromide was evolved. By reerystal- lising the reaction product from methyl alcohol, needles containing 1 mol. of the latter are formed. The melting point of this substance, dried at 100’, was 184°, (corr. 185°). Found: I me he EV MANGEN WEP Cale.’ for Le ENE, CG 72.14 72.30 71.90 H 10.36 10:07 CARIUS LIEBIG 0:55 Br 1448 14.50 15.40 15.07 1467 15.45 [e]Jp = + 3°,8 in chloroform. Most probably, «a monosubstitution product had formed and I now tried to obtain an additive product of the benzoate. When dissolved in a mixture of glacial acetic acid and carbon disulphide and then treated with a solution of bromine in glacial acetic acid, it yielded, after spontaneous evaporation of the carbon disulphide, beautiful leaflets. On extracting this product with boiling acetone a less easily soluble substance was left, which proved to bea monobromide. After repeated recrystallisation from aethyl acetate, I obtained fine, thick crystals which when melting were decomposed. Placed in the bath at 240° it melted at 2435. Found I II Lies. AV VO bs Wiss Vee yet IX X C 72.62 72.90 72.58 72.46 72.59 | H 8.85 8.88 8.72 9.09 8.84 CARIUS LIEBIG Br 13.14 13.04 12.97 13.40 13.01 Pacuined Cs HO Br -C = 1930, H=S6t, Br=12.87. [a]Jp = + 44°,9 in chloroform. The bromine atom is contained in the lupeol nucleus, because on saponification an alcohol containing bromine, and benzoicacid are formed. The more readily soluble portion crystallises from acetone in beautiful leaflets. It is also a monobromide but could not with certainty be characterised as a chemical individual. One of the means to trace the structure of a substance is the gradual destruction by oxidation. ( 468 ) The lupeol was, therefore, oxidised with the Kimani mixture’). Lupeol dissolved in benzene was shaken with a weighed quantity of the oxidising liquid, 6 atoms of oxygen caleulated for 1 mol. of lupeol. Titrations of the oxidising liquid with potassium iodide and sodium thiosulphate showed, that after six hours one atom of oxygen had been consumed and as the amount of chromic acid did not diminish any further, this one atom had been taken up quantitatively. The oxidation product, which crystallised from alcohol in beautiful, thick needles, melted at 169° (corr. 170°) and proved to be a ketone, to which I gave the name of lupeon. Found C 84.95 84.91 85.07 84.76 Cale. for C,, HO 85.24 H 11.64 11.81 11.62 11.61 11.59 11.09 laln = + 63°,1 in chloroform. Dr. JArGeER was kind enough to examine the crystalform of the lupeon. It belongs to the rhombo-bipyramidal class. A complete description will appear elsewhere. With hydroxylamine an oxime of the lupeon was obtained, which is but little soluble in alcohol. Reerystallised from ethyl acetate, it forms white, soft, light needles, which are decomposed when melting. Placed in the bath at 278°, they melt at 278°,5. Found C 81.98 Cale. for C,,H,, NOH 82.41 H 11.44 with lead chromate 10.94 N 308 3.11 [«|p = + 20°,5 in chloroform. Bromine dissolved in glacial acetic acid added to a solution of Inpeon in the same solvent gave hydrogen bromide and a dibromide, which was deposited from the acid. Reerystallised from a mixture of benzene and glacial acetic acid it consisted of beautiful, hard needles, which were decomposed when melting. Placed in the bath at 253° the melting point was 254’. Found 1 En Il Ev V Vi Vio Vib ie C 62.31 62.71 62.50 62.30 He “Sas 8.26 8.05 8.06 CARIUS LIEBIG nn en A Br 26.88 26.91 27.08 26.85 27.35 Aa Calc. for CHO Bes C= 6258; == ae Br = 26.90. [@]p = + 21°,4 in chloroform. When dissolved in ether, lupeon gave with hydrogen cyanide under the Se of a trace of ammonia a eyanohydrin, which Bee 34. 3564 (1901), ( 469 ) after some time deposited in the form of beautiful, thick needles. This substance is decomposed at a higher temperature and also on melting. Placed in the bath at 192°, it melts at 194°. By collecting the hydrogen cyanide liberated on heating in aqueous potassium hydroxide and then titrating with silver nitrate I determined the nitrogen content. Found: I II LTF: EY, ¥ VL MIE eale. for ‘C,2 B ON C 82.63 82.76 82.86 H 11.25 11.26 copper oxide lead chromate _ titrated 10.66 N 3.923.94 3.30 2.87°2.70 3.03 One mol. of cyanohydrin gave, with one mol. of ethyl alcohol and one mol. of hydrogen chloride, a substance, which, when placed in the bath at 230°, melted at 235°; as shown by a combustion, this was not, however, the expected ethyl ester of the corresponding acid. This substance has not been investigated further. Lupeol benzoate treated in the same manner as lupeol with the Kirranr mixture was not affected. Lupeon dissolved in benzene and stirred with the mixture for four hours at 40° also remained unaltered. By the action of chromic anhydride on lupeon at a higher tem- perature, acid products were formed, which could not be obtained in a crystalline state. The neutral oxidation product of lupeol with potassium perman- ganate and sulphurie acid consisted of a mixture, which could be separated only with extreme difficulty. Excepting lupeon no well- defined substance could be isolated from it. As SenKowskr') had obtained phthalic acid from cholic acid by oxidation with alkaline permanganate, I treated 23 grams of lupeol in the same manner, but it suffered complete destruction. This fact does, therefore, not favour the idea of a benzene nucleus in lupeol. By the oxidation of an acetic acid solution of lupeol acetate with chromic acid, I obtained a product which, on analysis, gave figures which agree satisfactorily with the calculated values for C,, H,, O,. Placed in the bath at 285° it melted at 295° to a dark brown mass. In alcoholic solution this substance did not turn blue litmus red, not even on diluting with water, but still it could be titrated very readily with alcoholic potassium hydroxide, phenolphtalein being used as an indicator. Assuming that one mol. consumes one mol. of KOH the titrations pointed to a molecular weight of 521 and 524, the formula C,, H,, O, representing 512,5. Found: C 77.59 77.23 76.87 77.24 calculat. for C,, H,, O, 77.28 H 10.75 10.49 10.09 10.79 — 10.23 ~ 1) Monatsh. f. Chem. 17. 1 (1896). ( 470 ) On saponification with alcoholic potassium hydroxide a substance was obtained which crystallised from ether in needles. Placed in the bath at 260°, the melting point was 263—265°. In regard to litmus this substance behaves like the unsaponified product, but it may be again titrated with alcoholic potassium hydroxide and phenolphthalein. From these titrations the molecular weight was found to be 452 and 461; the formula C,, H,, O, represents 470,5. Found: C 78.42 78.61 calculated for C,, H,, O, 79.08 Hed 07-11.05 10:74 The potassium compound of this substance is soluble, with diffi- culty, in aleohol, and erystallises from this in needles. On treating either the saponified or the unsaponified oxidation product the same compound was obtained, which seems to be a diacetylated substance. The results of the combustions, however, were not very concordant, but I have not been able to account for this. Found: C 75.39 74.71 75.67 74.96 74.47 calcul. for C,, H,, O, 75.75 H 10.12 10.16 10.51 10.24 9.81 By boiling with excess of aleoholic potassium hydroxide and titrating with aleoholie sulphuric acid the molecular weight was found to be 549, assuming that the molecule contains two acetyl groups. The formula C,, H,, O; represents 554.5, It is desirable to investigate more closely these oxidation products, which are so important in the study of lupeol, before trying to explain their formation. Lupeol is not reduced by metallic sodium and boiling amy! alcohol; whereas lupeon is reduced by sodium and ethyl aleohol to lupeol. Therefore, if lupeon should possess a double bond, this is sure not to be in @ 3-position in regard to the carbonyl group. Neither lupeol, nor lupeol acetate dissolved in boiling acetone are acted upon by potassium permanganate. This behaviour does not agree with the theory of a double bond, but the presence of the latter in lupeol and lupeon could be satisfactorily demonstrated by means of Hisr’s iodine reagent. On the other hand the oxidation product C©,, H,, O, no longer seemed to contain the double bond. On the strength of various combustions and bromine determinations, particularly of dibromolupeon, I consider C,, H,, O to be the most likely formula for lupeol. The formula C,, H,, O given by Likimrntk *) and Sack ®) is certainly not correct. Utrecht, Org. Chem. Lab. University. 1) LikieRNIK |. c. 2) Sack l, c, eve Chemistry. — “On a- and p-amyrin from bresk” 5. By Dr. N.H. Conen. (Communicated by Prof. Van Romeuren), (Communicated in the meeting of November 24, 1906). Communications as to g-amyrin, which is present as acetate in “bresk” or ‘‘djelutung’” have already been presented (These Proc. 1905, p. 544). Since then, I have prepared also g-amyrin cinnamate. This erystallises from acetone in small needles, which melt at 236,°5 (corr. 241°). In addition to g-amyrin and lupeol another substance was obtained from “bresk”, which proved to be identical with the e-amyrin found by VESTERBERG. This substance crystallises from alcohol in long, slender needles; m.p. 185° (corr. 186°). VusrerBrre gives the melting point as 181— 18125. Found: C 84.22 8430 calculated for C,,H,,O 84.43 H, 11.91 12.02 11.82 These, like all subsequent combustions have been made with lead chromate. [@]p =+82°,6 in chloroform; in benzene was found [+] —-+88°,2.”). For the purpose of characterisation, different esters were prepared from e-amyrin. a-Amyrin acetate was obtained by heating with acetic anhydride and sodium acetate. Recrystallised from aleohol it forms needle- shaped leaflets; m.p. 220—221°, (corr. 224—225°). VesSTERBERG gives the melting point as 221°. Found: C 81.85 82.27 81.79, calculated for C,,H,,0, 81.98 H 11.34 11.40 11.33 11.19 lalp = + 75°,8 in chloroform. a-Amyrin benzoate was obtained with the aid of benzoyl chloride and pyridine. From acetone it crystallised in long, prismatic needles ; m.p. 192°, (corr. 195°). According to VrsTrRBERG it melts at 192°. a-Amyrin cinnamate, which has not yet been described was obtained like the benzoate. When recrystallised repeatedly from acetone it forms small hard needles which melt at 176,5—177°, (corr. 178°). Utrecht. Org. Chem. Lab. Univ. 1) For a more elaborate description see, Diss. N. H. Conen. 1906, Utrecht, 2) VesrerBerG found in benzene [z]p = + 91°,6. Chemistry. — “On substances, which possess more than one stable liquid state, and on the phenomena observed in anisotropous liquids.” By Dr. F. M. Jararr. (Communicated by Prof. FRANCHIMONT). $ 1. The compounds now investigated belong to the series of fatty cholesterol-esters, which were the subject of a recent communication *). They are intended to supplement the number of the synthetic esters, studied previously and include: Cholesterol-Heptylate, Nonylate, Laurate, Myristate, Palmitate and Stearate. The Palmitic ester, as is well known, is also important from a physiological point of view, as it occurs constantly in blood-serum accompanied by the Oleate m.p. (43° C.) °). I have prepared these compounds by melting together equal parts by weight of pure cholesterol and fatty acid, and purifying by frac- tional crystallisation from mixtures of ether and alcohol, or ethyl acetate and ether. The details will be published later on in a more elaborate paper in the “Recueil”. The substances were regarded as pure, when their characteristic temperature-limits and the typical — transformations occurring therein, remained the same in every parti- cular, even after another reerystallisation, whilst also the solid phase, when examined microscopically, did not appear to contain any heterogenous components. Most of these esters were obtained in the form of very flexible, tabular crystals of great lustre and resembling fish-scales; some of them, such as the heptylate and the /aurate, erystallise in long, hard needles. The investigation showed, that most of these esters of the higher fatty acids possess three stable liqued phases. Whereas, in the first terms of the series one at least of these anisotropous phases was labile in regard to the isotropous fusion, all three are now stable under the existing circumstances, although sometimes definite, irre- versible transitions may still occur. It is a remarkable fact, that the stearate again exhibits an analogy with the lower terms, as it appears that only labile liguid-anisotropous phases may occur, or else none at all. A relation and similarity between the initial and final terms 1) F. M. Jaecer, These Proc. 1906; Rec. d. Trav. d. Chim. d. Pays-Bas, T. XXIV, p. 334—351. 2) K. Hürrure, Z. f. physiol. Chem. 21. 331. (1895); The blood serums of: man, horse, ox, sheep, hog and dog were investigated. ( 473 ) of the homologous series is plainly visible here. In what follows there will be described, tirstly, the thermic, and then the microscopic behaviour of these substances. § 2. The Thermometric Behaviour of these Substances. Cholesterol- Laurate exhibits the following phenomena : The isotropous fusion L of this substance has still, at 100° the consistency of gly- cerol, and gradually thickens on cooling. At 87°.8 C. (=¢,) there suddenly occurs a peculiar violet and green opalescence of the phase, which commencing at the surface, soon embraces the whole phase. The still transparent thin- jelly-like mass quite resembles a coagulating colloida! solution; the opalescence is analogous to that often noticed in the separation of two liquid layers. As the cooling proceeds, the opalescence colours disappear and the mass gradually becomes less transparent and also more liquid. It is then even thinner than the isotropous fusion Z. This doubly-refracting liquid A now solidifies at 82°.2 C. (=t,) to a crystalline mass S, accompanied by a distinct heat effect. If, however we start with the solid phase S and subject the same to fusion, the behaviour is apparently quite different. The substance softens and yields after some time a thick doubly-refracting mass, which will prove to be identical with the phase A. On heating further the viscosity decreases, and at about 86° it becomes very slight. There is, however, no sign of opalescence this time. The turbid mass may be heated to over 90°, without becoming clear and now and then A seems as if solid particles are floating in the liquid phase. At 90°.6 C. (= ¢,) everything passes into the isotropous fused mass LZ. The micros- copical investigation shows, that between A and Z another stable, less powerfully refracting liquid phase B is now traversed, and that, owing to retardation occurring, the phase S may be kept for afew moments adjacent to £, when A and B have already disappeared. This is therefore, a case where a substance may be heated a few deyrees above its actual melting point without melting. It should, however, be observed that the order of the temperatures is here quite irreconcilable with the phenomena considered possible up to the present, with homogenous substances; the temperature of 90°.6, at which these crystals disappear in contact with £ finds no place in the p-t-diagram of Fig. 1. Such a position of the said temperatures might be possible, when the system could be regarded as containing two components, for instance, if there was question of tautomeric forms which are transformed into each other with finite velocities. I think it highly probable that in all these substances, ( 474 ) “phenomena of retardation” play a great rôle ; moreover the enormous undercooling which the phase A can undergo without transformation, proves this satisfactorily in the majority of these esters. The different behaviour of the laurate on melting and on cooling the fused mass is so characteristic, that no doubt can be entertained as to the irreversibility of each series of transformations. Fuller details will be given below in the micro-physical investigation. § 3. Cholesterol-Nonylate forms at 90° an isotropous fused mass of the consistency of paraffin oil; on cooling to 89°5 a stable, greyish, doubly-refracting liquid appears which, gradually thickening, passes into a second strongly doubly-refracting liquid phase A, — which trans- formation is accompanied with a brilliant display of colours. All three liquids are, however, quite stable within each specific temperature- traject. On melting, as well as on cooling the substance, they succeed each other in the proper order. The viscous, strongly doubly-refracting, liquid phase A now becomes more viscous on cooling, and is finally transformed into a horny, transparent mass which exhibits no trace of crystallisation. Even after some hours, the often still very tenacious mass has not got crystallised. In the case of this substance it is therefore impos- sible to give the solidifying point or the exact temperature at which the heated mass begins to melt. The reason of this is, that the doubly refracting liquid A can be undercooled enormously and passes gra- dually into the solid condition without crystallising. As the micro-physical research has shown, a_ spherolite-formation occurs afterwards suddenly in the mass, which ultimately leads to the complete crystallisation of the substance. The velocity, with which such spherolites are formed appeared in some cases not to exceed 0.000035 0.000070 m.m. per second! § 4. Cholesterol-Myristate, at 80°, is still an isotropous, paraffin oil-like liquid. On cooling, it gradually becomes viscous; at about 82.°6 the glycerol-like phase then turns, with violet-blue opalescence, into a thick, strongly doubly-refracting mass A which, gradually assuming a thicker consistency, is finally converted into a horny mass, without any indication of a definite solidifying point. In this respect the substance is quite analogous to the previous one. On the other hand, on being melted, it behaves more like the laurate, in so far as it is converted into a double-refracting liquid 2, before passing completely into L. The transition temperature cannot be determined sharply, but I estimate it at about 80°, § 5. Cholesterol-Palmitate at 80° is a clear, isotropous liquid as thick as simple syrup. On cooling, the isotropous phase is con- verted at 80° with green opalescence into a fairly clear, transparent, doubly-refracting jelly A, which rapidly assumes a thinner consistency, and becomes at the same time more turbid, and finally solidifies at 77.°2, with a perceptibte caloric effect, to a crystalline mass S. In this case also, a doubly-refracting phase B appears to be traversed when the mass is being melted, before the occurrence of the isotropous fusion Z; I estimate the transition temperature at about 78°. § 6. With Cholesterol-Stearate, I did not succeed in demonstrating the occurrence of a doubly refracting liquid. The isotropous, thick- fluid fusion solidifies at 81° to well-formed crystals S. § 7. Cholesterol-Heptylate exhibits, in undercooled fusion only, one doubly-refracting liquid phase which is labile in regard to the solid phase S. The compound behaves, thermically, analogously to the caprylate. The temperature of solidification is at 110.°5, the tran- sition-temperature of the labile doubly-refracting phase lies a little lower. Of Cholésterol-Arachate, U could only obtain an impure product on which no further communications will be made. The ester could not be purified properly as it is not soluble to any extent in the ordinary solvents. The crude substance obtained does not seem to exhibit any anisotropous liquid phases. $ 8. Micro-physical behaviour of these substances. If a little of the pure solid cholesterol-laurate is melted on an object glass to an isotropous, clear liquid £, and the same is allowed to cool very slowly, there is formed, usually, a very strongly doubly- refracting, liquid phase, gleaming with lucid interference colours. It consists of large, globular drops, which exhibit the black axial cross and, on alternate heating and cooling, readily amalgamate to a syrupy, highly coloured, but mainly yellowish-white liquid. This phase will be called A in future. On cooling, it gradually thickens, until no more movement of the mass is noticed, which continues to exhibit a granular structure. Around this mass an isotropous border liquid is found. At first I felt inelined to look upon this tenaceous, isotropous mass, which is visibly different from the fusion L, as a distinct phase differing from the fusion A. But on using a covering glass and pressing the same with a pair of pincers, or by stirring with a very thin platinum wire, I found that this border liquid is only ( 476 ) ‘“pseudo-isotropous” (LEHMANN) and is, in reality, not different from A; only, the optical axes of the liquid crystals are all directed perpendicularly to the glass-surface. The other cholesterol-esters also exhibit this phenomenon. On further cooling, this phase A crystal- lises like the pseudo-isotropous border to a similar mostly spherolitic crystalline mass $. Between the spherolites one often sees currents of the pseudo- isotropous border liquid. If now the entire mass is allowed to solidify to S, and then again is melted carefully, it is at once transformed into the liquid A, recog- nisable by its high interference colours and its slow currents. Then, there appears suddenly a new, greyish liquid 4, consisting of smaller individuals with a less powerful doubie refraction, which after a short time is replaced suddenly by the isotropous fusion Z. If L is now cooled again, it is A which appears at once and not the phase B, Only a very feeble, greyish flash of light, lasting only for a moment, points to a rapid passing of the phase 5; it cannot, however, be completely realised now. On further cooling, S is formed sud- denly, sometimes in plate-like crystals. When once crystallisation has set in, S will not melt when the mass is heated, as might have been expected, but actually increase in size of the crystals occurs, and the velocity of crystallisation is now many, times increased. It must be remarked that the growing flat needles of S drive before them, at their borders, the liquid phase A amid violent currents. If the heating is now continued a little longer, we may notice sometimes, that whilst the little plates of S remain partly in existence, A passes first into the grey phase 4, which then is converted into the isotropous mass JL. We then have adjacent to ZL the solid phase S, which therefore, may be heated above its melting point, before disappearing finally into the isotropous fusion ZL. . All this shows, that the /aurate possesses three stable liquid phases and also that the isotropous fusion being coled, B is always passed over, but is realised when the solid phase is heated. All this is repre- sented in the annexed p-t-diagram; the arrows, therefore, indicate the order of the phases traversed on melting and on cooling. The phase A in its quasiimmovable period may be kept a long time solid at the temperature of the room, and may be considerably under- cooled before it passes into S. Notwithstanding its apparently solid appearance in that undercooled condition, A is still a tenaceous, thick liquid, as I could prove by stirring the mass with a thin platinum wire. The point (¢,) agrees with the opalescence which -occurs when Fig. 1. Schematic p-t-diagram for Cholesterol-Laurate. the isotropous fusion is cooled; this indicates, therefore, the moment where the stable phase B is replaced by the as yet still less stable phase A, which will soon afterwards be the more stable one; a fact which may be perhaps important in the future for the explanation of the analogous phenomena observed in the separa- tion of two liquid layers and the coagulation of a colloidal solution. Indeed, the transition at (f,) presents quite the aspect of a gela- tinising colloidal solution. The temperature of this transition point may be determined, but not sharply, at 87°.8. The temperature at which, when the solid substance melts, the liquid may be still kept turbid, owing probably to the presence of the meta-stable plate- like erystals of S, was determined at 90°.6; the solidifying tem- perature (¢,) lies at 82°.2. 32 Proceedings Royal Acad. Amsterdam. Vol. IX. ( 478 ) That the border liquid, obtained by cooling the isotropous fusion ZL, differed from ZL itself, could be demonstrated in more way than one. By heating and cooling we may get so far that, apparently, nothing more of A is visible, but that we have only the border liquid, which on cooling, crystallises immediately to S. Occasionally, the doubly- refracting individuals of A turn up in the mass for a moment to disappear again immediately. However, that isotropous liquid thus obtained is nothing else but A itself, when owing to the temperature variations, all individuals have, like magnets, placed themselves parallel with their (optical) axes and the whole has, consequently, become pseudo-isotropous. This same phenomenon also occurs with the other esters, for instance very beautifully with the nonylate and the myristate. The difference between these pseudo-isotropous phases and the isotropous fused masses / of these substances, is shown by the fact that the pseudo-isomorphous mass of A, and also the doubly- refracting portion of the same has a very thick-fluid consistency ; the isotropous fusion of the laurate has a consistency more like that of glycerol. As regards the solid phase and its transformation into the liquid condi- tion, it cannot be proved in this case that there exists a continuous transformation between the last solid partic lesand the first anisotropous ones. From the velocity, with which the diverse phases usually make room for each other in the microscopic examination, one would feel inclined to believe just the opposite. The thermic observation of the transformation, which generally exhibit only insignificant caloric effects, would, however, make the observer feel more inclined to look upon the matter as an uninterrupted concatenation of more or less stable intermediate conditions, which I have observed pre- viously with cholesterol-cinmamylate. A somewhat considerable heat effect occurs in some cases in the crystallisation of the solid phase only; in all other phases the exact transition temperature cannot be determined accurately by the thermic method. § 9. Cholesterol-Nonylate exhibits microscopically the following phenomena: Starting from the crystallised substance, this was fused first on an object glass to an isotropous liquid £. On cooling a greyish doubly- refracting liquid phase ZB appears, which, at a lower temperature, makes room for a very tenaceous, strongly doubly-refracting, mostly vellowish-white phase, A. This phase A is often surrounded by an isotropous border; if pressure is applied to the covering glass or if the mass is stirred with a very thin platinum wire, this isotropous liquid appears to be identical with A, and to be pseudo-isotropous by homoeotropism only. The optical axes of the doubly-refracting modification A again place themselves perpendicularly on the surface of the covering glass. On continued cooling A becomes increasingly thicker; at last a movement in the mass can be seen only on stirring. After a longer time there are formed from numerous centres in this tenacious mass thin, radiated spherolites, whose velocity of growth is but very small. When a number of these spherolites have formed and the mass is then heated carefully, the spherolites do not melt, but actually crease owing to the greater crystallisation-velocity. Soon afterwards —- however, they melt, on further heating, to the doubly-refracting phase A, where the circumferences of the spherolites and the black aaial crosses are preserved for some time; so that the whole much resembles a liquid mozaic. Subsequently the phase 5 reappears and afterwards the isotropous fusion £. The whole series of phases is traversed in a reversible manner; the liquid phase A, however may be so much undercooled, that a proper melting or solidifying point of the substance cannot be given. In larger quantities of the substance, the crystallisation does not set in till after some hours, and the substance turns first to a horny mass, which always remains doubly-refracting to finally exhibit local, white spots, from which the spherolite-formation slowly spreads through the entire mass. One would feel inclined to call this transformation of liquid- anisotropous into crystallised substance a continuous one, if it were only possible to observe, even for a moment, the intermediate con- ditions in that transition. As the matter cannot be settled by direct experiment, the transition must be put down, provisionally as a discontinuous one. In this case also, and the same applies to the other cholesterol- esters as well, the spherolite-structure of the solid phase is of great importance for this entire transformation of undercooled, anisotropous- liquid condition into the solid one. At the end of this communication I will allude briefly to a few cases from which the particular signi- ficance of the spherolite-structure in the transitions between aniso- tropous-liquid and anisotropous solid phases is shown also plainly in a different manner. $ 10. Cholesterol-Palmitate behaves in quite an analogous manner : I observed one solid phase and three liquid conditions A, B and L; as in the case of the laurate, B is generally observed only on warming. The succession of the liquid and solid phases takes place, however comparatively rapidly, so that a real solidifying point may 32* ( 480 ) be observed, which has also been proved by the thermic research. The solid phase crystallises in broad flat needles, when fused and then solidified in conglomerated spherolites. On melting, the thick, doubly-refracting liquid A is mostly orientated in regard to the previous solid spherolites. § 11. Cholesterol-Stearate could not be obtained in a doubly- refracting liquid form: the isotropous fusion always crystallises immediately amid rapid, rotating movements, to small needles, which often consist of a conglomeration of rosettes. It is possible that labile anisotropous phases are formed, owing to strong undercooling or by addition of some admixture *). §12. Cholesterol-Myristiate lends itself splendidly to the experiment. It behaves mainly in the same manner as the dawrate; the phase 5 can only be observed on heating, but not on cooling the isotropous fusion £. Most brilliant is the formation of large, globular erystal- drops of the modification A, also the colour-zone which precedes the formation of A from £, on cooling. This phase A also exhibits the phenomenon of pseudo-isotropism in a particularly distinct form. On the other hand, an important difference between this compound and the laurate is the much smaller velocity with which, on cooling, the spherolites S are formed from A; in this respect the compound exhibits more similarity with the nonylate. Sometimes it may be observed readily how in the phase A, which consists of an enormous number of linked, globular crystal-drops, which all exhibit the black cross of the spherolite crystals, centrifugal current-lines are developed from a number of points in the mass, along which the crystal-drops range themselves. After the lapse of some time those doubly-refracting globules are seen to disappear, while the current-lines have now become rays of the spherolite. Here again, the question arises whether the transformation of the doubly-refracting liquid globules, which are orientated along the current-lines, into the true spherolite form, does not take place continuously, and whether we do not speak of a sudden transformation merely because we are not able to observe the stadia traversed in this transformation. The liquid globules of the phase A themselves exhibit much similarity with a kind of liquid spherolites; a few times, | have even been able to observe such “liquid spherolites” of greater dimens- 1) Prof. LenMANN informed me recently that the stearate possesses indeed two labile, anisotropous liquid phases. ( 481 ) ions, which rapidly solidified to solid spherolite crystals. In the case of this compound also, one feels convinced that there must exist a very intimate relation between the spherolite-formation of a substance and its power of forming anisotropous liquid phases; on the nature of this relation, I hope to make a communication later on. It may, however, be observed, provisionably that in all trans- formations: liquid S solid, where serious “phenomena of retardation” may occur, the undereooling, or superfusion, for instance is generally abrogated amid a differentiation of the phase into spherolites. All the cholesterol-derivatives, mentioned in this paper, exhibit this sphe- rolite-formation. In the case of a-phytosterol-propionate, 1 have been able to show, that a complex of a large number of doubly-refracting microscopic spherolites may imitate the optical peculiarities of the liquid phases in process of separation and of the colloidal opalescence. This might lead to the strengthening of the previous conception of the colloidal solidification as a separation-phenomenon of labile liquids. § 13. Cholesterol-Heptylate contains only labile liquid anisotropous phases. It exhibits great similarity with the caprylate described previ- ously: I have only a few times been able to obtain one single thick- fluid phase A from the undercooled isotropous fusion L. The solid phase erystallises rapidly and in beautiful flat needles, which exhibit high interference colours. On warming, the substance readily migrates towards the colder parts of the object glass. § 14. In conelusion, I will communicate a few more points as regards some phenomena, which prove plainly the significance of the spherolite structure for with these questions. Some time ago, I published a research on the fatty esters from Phytosterol from Calabar-fat and stated how they all are wont to erystallise in the spherolite-form from their cooled, isotropous fused mass, while anisotropous liquid phases are not observed therein, with the exception of the normal valerate which possesses a thick-fluid anisotropous modification, and exhibits the phenomenon of the chang- eable melting point, which again becomes normal on long keeping; a fact also observed in the case of a few fatty glycerol-esters. Since then, Winpaus has proved that the phytosterol, extracted from Calabar fat is a mixture of two isomorphous phytosterols, which cannot be separated by crystallisation. Being engaged in preparing the pure fatty esters from the principal of those two phytosterols, namely the «-compound (m.p. 136°), I discovered that the fused propionate of a-phytosterol (m.p. 108°), when cooled rapidly in cold water, ( 482 ) exhibited the most brilliant interference-colours, which is also the case with the cholesterol esters (acetate for instance), which possess labile anisotropous liquid phases. The thought naturally at once occurred, to attribute these phenomena to the appearance of liquid crystals in the now pure a-phytosterol-ester. A similar behaviour was also shown by perfectly pure a-phytosterol-acetate, but with a much less display of colours. It was, however, a remarkable fact, that a-phyto- sterol-propionate even after complete solidification still retained those colours for an indefinite length of time, particularly at those sides of the testtube, where the layer of the substance was thinner and had cooled rapidly. The microscopic investigation now showed that these two sub- stances exhibit extremely rapidly disappearing anisotropous liquid phases or, more probably, none at all"); but that the said colour-phenomenon is caused by a very peculiar spherolite-structure. In what follows, I have given the description of the solidifying phenomena of the «-propionate, and also a figure representing the typical structure of the fused and then cooled compound, such as is present at the coloured sides of the tube. If a little of the solid substance is fused on a slide to an isotropous liquid the fol- lowing will be noticed on cooling. The mass solidifies completely to spherolites, namely to a conglomeration of circular, concentrically grouped figures, which appear connected with a series of girdles. When three spherolites meet, they are joined by means of straight lines which inclose angles of about 120°. The mass is slightly doubly-refracting and of a greyish colour; the rings and girdles are light greyish on a darker back-ground. Each spherolite exhibits besides a concentric structure, the black cross, but generally very faint. The whole resembles a drawing of polished. malachite from the Oeral, or of some polished agates. 1) Whereas the phytosterol-esters from Calabar fat which, of course, contain a definite amount of the 6-homologue, exhibit mo liquid crystals, the pure z-esters commencing with the butyrate [or perhaps the propionate] did show this pheno- menon. This discovery is a powerful argument against the remarks often made in regard to the cholesterol-esters, that the remarkable phenomena described are attributable to an admixture of homologous cholesterols. Foreign admixtures prevent as a rule these phenomena altogether; in any case they are rather spoiled than improved. ( 483 ) The walls of the test-tube or the object-glass, which exhibit the said colour-phenomena, have that same structure, but with this diffe- rence, that the globular, concentrically deposited spherolites have much smaller dimensions and lie much closer together. Each little spherolite has also a cross; this however, is not dark, but coloured with yellow and violet arms. The spherolite is also coloured in the alternate circle-quadrants. This ensemble of small, coloured spherolites is the cause of the said brilliant colour-phenomena; they are quite analogous to those which are wont to appear in the case of liquid crystals and remain in existence for an indefinite period. Each of them exhibits one or generally two luminous points in the centre; they exhibit a strong circular polarisation and are left-handed. The whole appears between crossed nicols as a splendid variegated mozaic of coloured cellular parts. The size of each individual is 0.5-——1 micron. The acetate also exhibits something similar, but the spherolites are built more radial and the whole is not at all so distinct. I hope to contribute more particulars as to these remarkable phytosterol-compounds shortly. I have mentioned them here merely to show the importance of this structure-form for the optical pheno- mena, observed in the anisotropous phases. Zaandam, 14 November 1906. Chemistry. — “On irreversible phase-transitions in substances which may exhibit more than one liquid condition.” By Dr. F. M. JAEGER. (Communicated by Prof. FRANCHIMONT). (Communicated in the meeting of November 24, 1906). § 1. The fatty esters of a-Phytosterol from Calabar-fat, which the Phytosterol mostly occurring in the vegetable kingdom, and which has also been isolated from rye and wheat under the name of “sitosterol”’, exhibit very remarkable properties in more than one respect. In my previous communication, I ailuded briefly to the colour phenomena and the spherolite-structure in the propionate and the acetate. In the latter [ could not observe anisotropous liquid phases ; n the former a doubly-refracting phase is discernible just before melting, but it lasts too short a time to allow the accurate measure- ment of the temperature-traject. With the following four terms of the series, however, these pheno- mena are more distinct, and occur under conditions so favourable as ( 484 ) could hardly be realised up to the present in the other known sub- stances. They also exhibit enormous phenomena of retardation in their diverse transitions and often a typical irreversibility thereof, of which L will now communicate some particulars. § 2. Thermometrical behaviour of the fatty «-phytosterol- esters. A. a-Phytosterol-n.-Butyrate, on very slowly raising the temperature, melts at 89°.5 to a turbid, doubly-refracting liquid A, which at first is very viscous but rapidly becomes thinner and is converted, at 90°.6, into a clear isotropous fusion £ of the consistency of glycerol. On cooling the same carefully, the thermometer falls gradually while the isotropous liquid thickens more and more but remains quite clear. At 80° the whole mass crystallises all at once to small crystals S with so great a caloric effect that the thermometer goes up to 85°. There is no question now of anisotropous liquid phases at all. These two experiments may be repeated at will but: always with the same result. As to the nature of the turbid phase, compare “micro-physical behaviour”. If the isotropous fusion is suddenly cooled in cold water, a bluish- grey coloration appears and a soft, doubly-refracting mass is obtained, which does not become crystalline until after a very long time. B. «-Phytosterol-Isobutyrate, when treated in the same manner, melts at 101°.4 to a glycerol-like, turbid, doubly-refracting liquid A, which gradually assumes the consistency of paraffin-oil and is con- verted at about 103°.2, apparently continuously, into a clear fusion L. If this is cooled, it certainly becomes gradually thicker but it still remains quite clear and isotropous. At 80°.4 it becomes turbid and doubly-refracting ; this phase is identical with A, and it has the consistency of glycerol; at 73° it has become as thick as butter, and at 66° the thermometer can be moved only with difficulty, whilst it may now be drawn into sticky, doubly-refracting threads. At 65° the thefmometer suddenly goes up to 68°.8 and the mass erystallises in long, delicate needles JS. On rapid cooling of the fused mass, this is converted into a turbid, greasy looking, doubly-refracting mass, which crystallises but very slowly; no colour-phenomena occur. C. a-Phytosterol-n.- Valerate melts, when in the crystallised condi- tion, at an wncertain temperature. At about 48°, the substance com- mences to soften visibly, at 54° its consistency is that of thick butter, at 80° it is somewhat thinner, at 85° it is actually liquid, but still turbid and doubly-refracting. A// these transformations proceed quite ( 485 ) continuously. At about 97.°5 the liquid is clear and isotropous; it has then the thickness of paraffin-oil. If, however, the isotropous fused mass is cooled, the isotropous paraffin-oil-like liquid remains clear to about 87.°3, when a turbid doubly-refracting phase is formed. This, on further cooling, gradually becomes more viscous; at 80° it is as thick as butter, at 66° it can hardly be stirred, and may be drawn into threads. It may be cooled to the temperature of the room without solidifying. It remains in this condition for hours, but after 24 hours it has again become crystalline. The substance, therefore, has no determinable melting or solidifying point. D. «-Phytosterol-Isovalerate behaves quite analogously to the n- valerate. Neither a definite melting point, nor a solidifying point can be observed. The mass softens at about 45°, is anisotropous thick- fluid at 65°, and becomes clear and isotropous at 81°. On cooling to 78.°1, a beginning of turbidity is noticed, the liquid gradually becomes thicker and is converted at an uncertain temperature into a tenacious sticky, doubly-refracting mass, which after 24 hours has again solidified to a crystalline mass. § 3. The thermometrical behaviour of these remarkable substances is represented in the annexed schematic p-f-diagram, for the case of the n-butyrate and isobutyrate. The typical irreversibility of these phenomena is thus seen at once. Moreover in the case of the two valerates, the whole behaviour can be described only as a real, gradual transformation, solid 2 liquid with an intermediate realisation of an indefinite number of optically-anisotropous liquids. $4. The micro-physical behaviour of the fatty u-phytosterol esters. Perhaps, there are no substances known, which exhibit under the microscope the characteristic phenomena of anisotropous liquids in so beautiful and singular a manner as these esters; in this respect the isobutyrate and the valerate excel in particular. In the normal butyrate, the traject, where the liquid crystals are capable of exis- tence is rather too small. For this reason, although the behaviour of the four substances differs in details, I will describe more particularly the behaviour of the „-valerate and as to the others, | will state occasionally in what respect they differ from the valerate. In conse- quence of the totally different circumstances which the microscopic method involves, nothing more is seen of the thermically observed peculiar irreversibility and even progression of the transformations. For the study of the nature of the diverse phase-transformations, the ( 486 ) Fig. 1. Schematic p-t-diagram for -Phytosteryl-Isobutyrate. Fig. 2. Schematic p-t-diagram for z-Phytosteryl--Butyrate. thermometric method is certainly preferable to the microscopic one, because in the latter, the delicate changes in temperature cannot be controlled so surely as in the first method. For this reason, the phase- ( 487 ) transformations, when observed microscopically, convey the impression of being more sudden than in the thermic observation. Still, the microscope completes the task of the thermometer in a manner not to be undervalued, at it gives an insight into the structure of the diverse phases and allows one to demonstrate their difference or their identity. § 5. If a little of the beautifully crystallised n-valerate is carefully melted on an object glass, the substance, at a definite temperature changes, apparently suddenly, into an aggregate of an enormous number of globular, very large and strongly doubly-refracting liquid- drops, which all exhibit the black cross of the spherolites ') but can flow really all the same. This condition may be rendered permanent for a long time at will. But they may also amalgamate afterwards to larger, plate-like, highly coloured liquid individuals, somewhat resembling sharply limited crystals. These are frequently multiplets of liquid drops; the demarcations between the separate individuals vary constantly by changes in temperature. The isotropous border of the mass is very striking. By pressure or by moving the covering glass, also by the sliding currents which we can induce herein by changes in temperature, it may be readily shown that this isotropous border, owing to a parallel orientation of the liquid individuals, is only psendo-isotropous and really identical with the rest of the phase. Sometimes one may succeed even in communicating this pseudo-isotropous aspect to the entire mass *) by 1) We can, however, often observe a slanting projection of the optical symmetry axis, which gives the same impression as if we look perpendicularly to one of the optical axes of a biaxial crystal, or on a monoxial crystal cut obliquely to the optical axis. We observe at the same time coloured rings which exhibit an elliptic form. It is very remarkable that, when the phase has become very viscous on cooling, these ellipsoidal drops, provided with rings and slanting but mutually parallel-directed axes may be kept for a long time in an apparently immobile conditicn in the midst of the pseudo-isotropous or double-refracting liquid. They place them- selves mutually like little ellipsoida: magnets. However, [ could observe, that these drops are often not quite ellipsoidal, but that they are sharply broken a little at the one side, just there, where the optical axis is slanting. By turning the object-table, the axial point turns in the same direction as the table, while the black line or cross is preserved. (Added in the English translation Januari 1907). *) The anisotropous-liquid phase has, in the case of the two valerates, an extra- ordinary tendency to place itself in this pseudo-isotropous condition. We can observe this, because the border of the drop often moves inward with widening of the isotropous-looking line. It is also remarkable to see how the flowing crystals when meeting an air bubble arrange themselves close together, normally on the border thereof. ( 488 ) often repeated warming followed by rapid cooling. This substance is about the best known example of this phenomenon. § 6. If now we go on heating very cautiously, the larger flowing crystals and also the smaller drops situated between them are seen to move about rapidly; the larger individuals, which consist mostly of twins or quadruplets, are split up into a multitude of globular drops and these, together with the smaller ones, disappear at a definite temperature entirely in the isotropous liquid, which is now isotropous in reality. The globules of the liquid rotate to the right and the left under distortion of the mass, as may be observed from the spiral- shaped transformation of the black cross. Sometimes, before the mass becomes isotropous we may notice a temporary aggrandisement of the plate-like flowing crystals at the expense of the smaller interjacent globules; a result of the momentarily increased crystallisation-velocity due to heating. § 7. On cooling the isotropous fusion this is first differentiated into an infinite number of the double-refracting liquid globulus, which here and there amalgamate to the more plate-like flowing crystals. On further cooling, these latter individuals remain in existence notwithstanding the undercooling, while the little globules in the meanwhile unite to the same kind of plate-like individuals. This aggre- gate, brilliant in higher interference colours becomes in course of time thicker and thicker in consistency while the aggregation, owing to an apparent splitting, becomes more and more finely granu- lated. But even after the lapse of some hours, the phase remains anisotropous-liquid as may be easily proved by shifting the mass and by the pseudo-isotropous border, which commences to exhibit delicate, double refracting current-lines. In the end when the pseudo-isotropous liquid has passed like the remainder into the same, almost completely immobile aggregation of doubly-refracting individuals, it is, gradually, transformed after a very long time into an aggregate of plates and spherolite-like masses, which possess a strong double refraction. § 8. If, after the lapse of some hours, the partially or completely solidified mass is melted cautiously, we sometimes succeed, in the case of the two valerates, in keeping the crystals of the phase S (therefore the solid crystals) for a few minutes near the isotropous fusion £ at a temperature above the highest transition point. This phenomenon is, therefore, again quite homologous to that first observed by me with cholesterol-laurate and which might be described as a ( 489 ) heating of a solid substance S above its melting point without fusion taking place. For the present, at least according to existing ideas, this behaviour can only be explained by assuming the presence of a two- component-system with tautomeric transformations subject to a strong retardation. When the isotropous fusion £ which has scarcely cooled to a few doubly-refracting drops is melted cautiously, we may observe some- times that where a moment before the strongly luminous, yellowish- white globules were visible, there are now present greyish globules showing the black cross, which gradually decrease in size and also darken, to disappear finally as (isotropous?) little globules in the isotropous fusion *). This phenomenon, in connection with those of crystallised ferric chloride to be described later, and with similar phenomena observed with the cholesterol esters appears to me to have great significance for the theory of the formation of liquid crystals. § 9. Finally, there is something to be observed as to the separation of «-Phytosterol-valerate from organic solvents. The substance may be obtained from ethyl acetate + a little alcohol in beautiful, hard, well-formed little crystals. If, however, the saturated co/d solution in ethyl acetate is mixed with much acetone (in which the substance is but sparingly soluble) the liquid suddenly becomes a milky-white emulsion which deposits the compound not as a fine powder, but in the form of a doubly-refracting, very thick and very sticky liquid. I have repeated this precipitation in a hollow object glass under the microscope. The emulsion consists of a very great number of doubly-refracting, globular liquid-globules, which are either moving about rapidly in the liquid, or, when united to larger masses, are quite identical with the ordinary anisotropous phase A, when this is cooled to the temperature of the room. These little globules all exhibit the cross of the spherolites, and the doubly-refracting liquids. They soon become solid and then form small needles and spherolitic aggregations. It may be easily proved by stirring that the globules deposited first are liquid; moreover, the doubly-refracting masses often communicate with each other by means of very narrow, doubly- refracting currents, while they often exhibit the phenomena of pseudo- isotropism. Therefore, we have evidently obtained here the liquid-anisotropous 1) Before that happens, we may sometimes see here the globules becoming enlarged to multiplets by amalgamation there larger ones being changed into smaller ones, sometimes here one disappearing in the liquid while very close by new individuals appear. ( 490 ) phase A from a solution by rapid precipitation at the temperature of the room, and that in isolated drops! A few other phytosterol esters exhibit analogous phenomena which | will describe later on in a more elaborate communication on these substances. § 10. A very remarkable fact in the n-valerate, the iso-valerate and the ssobutyrate, is the differentiation of the isotropous fusion into a large number of globular, doubly-refracting liquid drops of con- siderable dimensions, which like the circles of fat on soup float alongside and over each other and often unite to multiplets, whose separate parts are still recognisable. Wreathed aggregations of the liquid globules are also observed occasionally. In most cases the separate liquid globules exhibit the black cross and the four luminous quadrants grouped centrically. They are, however, also seen to roll about frequently, so that the projection of the optical symmetry axis now takes place excentrically. Owing to the enormous size of the individuals and the low temperature-limits, these esters lend them- selves to the study of these phenomena certainly VoRLÄNDER's p- azoxy benzoic-ethy lester. If the temperature of the mass, when totally differentiated into liquid globules — and the isobutyrate is particularly adapted for this differentiation — is slightly raised, the liquid globules are often seen to disappear suddenly just after they have enlarged their limits as it were by an expansion. It is like a soap-bubble bursting by over blowing. § 11. Finally, I wish to observe that the thermical transitions just described and particularly those of the two valerates, can only be interpreted by assuming a quite continuous progressive change. For all these gradual transformations. either on melting or on soli- difying, a measurable time is required and nowhere is to be found any indication of a sudden leap. An exception is, however, afforded by the sudden crystallisation of the two butyrates. § 12. As regards the differentiation of the fusion JZ into an aggregate of anisotropous liquid globules, I will now make a com- munication as to an experiment upon the erystallising of ferric chloride hexahydrate, which substance exhibits something similar, and which, like most undercooled fusions and like many compounds which exhibit liquid crystals, erystallises in typical spherolites. If we melt the compound Fe, Cl, + 12 H,O cautiously in a little tube, taking care that no water escapes, and a drop of this brownish- ( 491 ) red fusion is put on an object glass, it may be left for hours at the temperature of the room without a trace of crystallisation being noticed. The liquid is now greatly undercooled and exists in a state of metastable equilibrium. For all that, it has the same chemical composition as the solid phase from which it was formed. On prolonged exposure, small liquid globules appear locally in the fairly viscous mass, probably owing to local cooling, or by a spon- taneous evaporation of water at those points. These liquid globules are quite isotropous and are surrounded by a delicate aureole having an index of retraction different from that of the rest of the liquid (fig. 3a). The ob- servation shows that, optically, they are, practically, no denser than the liquid, and from the fact that they afterwards become, im their entirety, a Fig. 3a. spherolite of the hexahydrate, we must conclude that their chemical composi- tion does not differ from that of the fused mass. These globules of liquid are con- verted gradually into doubly-refracting masses whose section is that of a regular hexangle with rounded off angles; individual crystals are not yet visible in the doubly-refracting mass and the luminous zone around still appears to exist (fig. 34). Fig. 30. Here and there, hexangular, sharply limited, very small plate-shaped crystals are also seen to form in the liquid without previous formation of liquid globules"). In the end, the doubly- refracting hexangular mass gets gra- dually limited by more irregular sides, while a greater differentiation of the mass into light and dark portions points to a crystallisation process com- Fig. 3c. mencing and progressing slowly. 1) These may, however, be formed perhaps owing to the presence of traces of sal ammoniac, Ei ( 492 ) Finally, we can observe a spherolite of the hexaliydrate with a radial structure which now grows centrifugally to the large well- known semi-spheroidal spherolites of ferric chloride (fig. 3c). § 13. This experiment proves that the abrogation of the metastable condition, or at all events of a liquid condition which is possible under the influence of phenomena of retardation may happen owing to the formation of spherolites which are preceded by the differentiation of the fusion into an aggregate of liquid globules. True, the latter are here isotropous in contrast with the phytosterol esters just described, but the anisotropism of the latter liquids may be caused also by factors which are of secondary importance for the apparently existing connection between: metastability of liquid conditions, their abrogation by spherolite formation and the possible appearance of liquid globules as an intermediate phenomenon. I will just call atten- tion to the fact that if we set aside a solution to erystallise with addition of a substance which retards the crystallisation, this will commence with the separation of originally isotropous liquid globules, so-called globulites, which BrHrenps and VoGrrsane commenced to study long time ago. All this leads to the presumption that the formation of the aniso- tropous liquid phases as aggregates of doubly-refracting liquid globules may have its origin in a kind of phenomena of retardation, the nature of which is still unknown to us at the present. Before long, I hope to revert again to this question. Zaandam, 21 Nov. 1906. Physics. — “Some additional remarks on the quantity Hand MAxweLr’s distribution of velocities.’ By Dr. O. Postwa. (Communicated by Prof. H. A. Lorenz). § 1. In these proceedings of Jan. 27th 1906 occur some remarks by me under the title of: “Some remarks on the quantity H in BorrTzMaNN's Vorlesungen über Gastheorie”. My intention is now to add something to these remarks, more particularly in connection with GrBBs’ book on Statistical Mechanics 5), and a paper by Dr. C. H. Wind: “Zur Gastheorie” ’). In my above-mentioned paper I specially criticised the proofs given 1) J. Wittarp Gipss “Elementary Principles in Statistical Mechanics”, New-York, 1902. 2) Wien. Sitzungsber. Bd. 106, p. 21, Jan. 1897, ( 493 ) -by BorrzmMann and Jeans that Maxwerr’s distribution of velocities in a gas should give the most probable state, and demonstrated that they wrongly assume an equality of the probabilities a priori that the point of velocity of an arbitrary molecule would fall into an arbitrary element of the space. The question, however, may be raised whether it would not be possible to interpret the analysis given by BorrzMaNN and JEANS in a somewhat different way, so that avoiding the incorrect fundamental assumption, the result could all the same be retained. And then this proves really to be the case. When the most probable distribution of velocities is sought from the ensemble of equally possible combi- nations of velocities with equal total energy, we make only use of the fact that the different combinations of velocities are equally possible, how they have got to be so is after all of no consequence. Or else, it had not been necessary to occupy ourselves with the separate velocities of the molecules and make an assumption as to them. This way of looking upon the matter is of exactly the same nature as that constantly followed by Gisps in his above-mentioned work. GiBBs treats in his book all the time instead of a definite system, an ensemble of systems of the same nature and determined mostly by the same number of general coordinates and momenta (p,.. - Pn Yi: -++Qn); Which he follows in their general course. Such an ensemble will best illustrate the behaviour of a system (e.g. a gas-mass), of which only a few data are known and of which the others can assume all kinds of values. He calls such an ensemble micro-canonical when all systems, belonging to it, have an energy lying between /# and EJ dE and for the rest the systems are uniformly distributed over all possibilities of phase or uniformly distributed over the whole extension-in-phase the energy of which lies between / and E + dk. When the energy of a gas-mass is given (naturally only up to a certain degree of accuracy) we should have reason according to G1sBs to study the microcanonical ensemble determined by this energy, and to consider the gas-mass as taken at random from such an ensemble. The extension-in-phase considered is thought to be determined by fan ... dn, but in the case of a gas-mass with simple equal molecules this is proportional to js da, dy, de, … . dan dyn den, de, dy, de, … . dan dyn den, so that. we may say that every combination of velocities and con- figuration is of equally frequent occurrence in the ensemble. It is now easy to see that when the energy is purely kinetic the 33 Proceedings Royal Acad. Amsterdam. Vol. IX. ( 494 ) same cases occur in such an ensemble, with regard to the distribu- tion of velocities, as are considered as equally possible cases by Bo.TzMANN and Jeans. The difference in the way of treatment of Gisps on one side, and that of Bottzmann and Jeans on the other consists besides in the fact that the one occupies himself with separate velocities and the other not, in this that GrBBs treats the configuration and the distribution of velocities at the same time (both belong to the idea phase), whereas Juans treats the latter separately, and Borrz- MANN does not occupy bimself with the configuration in this connection. Every phase of BoLTZMANN (combination of velocities) corresponds with as many phases of GiBBs (combination of velocities and con- figuration) as the molecules can be placed in different ways with that special combination of velocities. This number being the same for every combination of velocities according to the independence of the distribution of velocities and configuration following from the fundamental assumption, it will be of no consequence, comparing the different combinations of velocities inter se, whether we also take the configuration of the molecules into account or not. So when seeking the most probable distribution of velocities (that, with which the most combinations of velocities coincide), we must arrive at the same result whether we follow GiBBs or BOLTZMANN. It is obvious that the phases of the microcanonical ensemble meant here are what GiBBs calls the specifie phases. GiBBs distinguishes namely between specific and generic phases: in the former we con- sider as different cases those where we find at the same place and with the same velocity, other, even though quite equal molecules, in the latter we do not. In other words: in the former we consider also the individual molecules, in the second only the number of the molecules. So we may now say that in such a microcanonical ensemble the most probable distribution of velocities and that which will also occur in the great majority of cases (compare JEANS’ analysis discussed in the first paper) will be that of Maxwerr. When therefore an arbitrary mass of gas in stationary state may be considered as taken at random out of such a microcanonic ensemble, Maxwerr’s distribution of velocities or one closely resembling it will most probably occur in it. In this way a derivation of the law has been obtained to which the original objection no longer applies, though, of course, the assumption of the microcanonical ensemble remains somewhat arbitrary *). 1) With the more general assumption of a canonical ensemble Maxwett’s law is derived by Lorentz: “Abhandlungen über Theoretische Physik”, Lpzg. 1906 I, p. 295, ( 495 ) Finally the question might be raised, when we want to consider the separate velocities, whether it is possible to arrive at the equally possible combinations under discussion on another suppo- sition a priori about the chances of every value for the velocity than the one indicated by BoLtTZMANN and Jeans. The supposition must of course be such, that the chance is independent of the direction of the velocity, so that the chance of a velocity c, at which the point of velocity falls into a certain element of volume d<d:,ds, may be repre- sented by f(-)dsdds. When we moreover assume that the probabilities for.the different molecules are independent of each other, the probability of a certain combination of velocities is proportional to f (c,) f (c,).….f (co), and this must remain the same when the kinetic energy Z, or because the molecules are assumed to be equal, Yc? remains the same. For every change of c, and cq into cz and c', so that eo; + cy =c*%, +c", must f(cr).f(c) =f (CH). fic). This is an equation which frequently occurs in the theory of gases, from which follows /(c) = ae’. As a special case follows from this: f(c) == a, i. e. the assumption of BorTZMANN and JrANs, that the probability a priori would be equal for every value of the velocity. $ 2. In the second place I wish to make some remarks in con- nection with the proof that BoLrzMann gives in his “Gastheory”, that for an ‘‘ungeordnetes” gas with simple suppositions on the nature of the molecules in the stationary state Maxwerr’s distribution of velocities is found. Dr. C. H. Wiyp shows in his above-mentioned paper that in this BOLTZMANN makes a mistake in the calculation of the number of collisions of opposite kind. Bo.tzmann, namely, assumes, that when molecules whose points of velocity lie in an element of volume dw, collide with others whose points of velocity lie in dw,, so that after the collision the former points lie in dw’ and the latter in dw,', now the elements of volume dw and dw’, dw, and dw,’ would be equal, so that now dw'dw', = dw dw. He further assumes that when molecules collide whose points of velocity lie in dw’ and dw’, they will be found in dw and dw, after the collision. These last collisions he calls collisions of opposite kind. WiNp now shows that this assumption is untrue; dw is not = dw’, dw, not =do',, nor even dwdw, = dw'dw,', except when the masses of the two colliding molecules are equal’). Further the points of velocity of colliding molecules which lay in dw’ and dw’, do not always get to dw and dw, after collision, 1 I point out here that even then it is not universally true, but only when the elements of volume dw and dw, have the shape of rectangular prisms or cylindres whose side or axis has the direction of the normal of collision. 33* ( 496 ) so that another definition is necessary for collisions of opposite kind, viz. such for which the points of velocity get in dw and dw, after the collision. Winxp proves further that the number of collisions of opposite kind is all the same represented by the expression which BOLTZMANN had found for it. It is then easy to change (what Wiunp does not do) the proof given by BOLTZMANN in $ 5 of his ‘Gastheory”, that Maxwell’s distribution of velocities is the only one possible, in such a way that it is perfectly correct. But the error in question makes itself felt all through BorrzmanN’s book. Already with the proof of the H-theorem given in more analytical form in a footnote to § 5 we have some difficulty in getting rid of this error. We meet the same thing when the molecules are treated as centres of force, and when they are treated as compound molecules. At the appearance of the second volume of his work, BorrTzMaNN had taken notice of Wunp’s views, but the inaccurate definition for collisions of opposite kind has been retained *). In connection with this error, made by BoLTZMANN in a geometrical treatment of the phenomena of collision, is another error of more analytical nature, so that also Jrans, who treats the matter more analytically, gives a derivation which in my opinion is not altogether correct. Though preferring the geometrical method, BoLTZMANN repeat- edly refers to the other’). The method would then consist in this, that the components of the velocities after the collision §'7$'§',7',$', are expressed by /(§1$§,7,5,) and then by means of Jacosi’s func- tional determinant d&'dy'd8d3',dy',d6',_ is expressed in d&dyddé,dy,d6, . We find then that here this determinant is =1 and so d§'dy/dedé',dy',d3', = d&dydtd&,dy,d8, or dw'dw', = dodo. The number of collisions of opposite kind = /'F",dw'do',o*g cos 9didt according to BortzMANN, and so also = f'F",dwdw,o’g cos 9dâdt. In this the mistake is made, however, that d&dnd8d8' d.d, the volume in the space of 6 dimensions that would correspond with the volume ddydéds,dy,d5, before the collision, is thought as bounded by planes such as §'=c, which is not the case. Jeans too equates’ the produets of the differentials, in which according to him, dg’. . . dS’, being arbitrary, the d§...d$ must be chosen in such a way, that the values of &'...6, calculated by the aid of the functions 8'—=f(&...5) ete. fall within the limits fixed by dé etc *). This, however, is impossible. 1) Cf § 78, 2nd paragraph. 2) Cf. among others volume I, p. 25 and 27. 3) Cf. *The dynamical Theory of Gases” p. 18.. ( 497 ) In my opinion the correct principle that the calculation of the extension occupied by the combinations of the points of velocity after the collision when that before the collision is known and vice versa, would come to the same thing as a transition to other vari- ables in an integration, has not been applied in exactly the correct way. The property in question says that in an integral with transi- yl tion from the variables §'7'$'§',7,5', to §7$§,7,$, the product of the | J ; SSS n ey differentials d&'dy'd9d3',dy',d0, may be replaced by zlib MENSE NS) dsdydSds,dy,d$,, if we integrate every time with respect to the corre- sponding regions, but these expressions are not equal for all that. The first expression may be said to represent the elementary volume in the space of 6 dimensions, bounded with regard to S...8,, the second the elementary volume bounded with regard to §...6, °). We have a simple example when in the space of three dimen- sions we replace f pdadydz, which e.g. represents the weight of a body, by | pr? sin 8drdddg, which represents the same thing, without de dy dz having to be equal to 7? sin 9 dr dd dg. So we have here: HEB je a fa du d8 d§', du’, dS, =f Tet ds dy dö d$, dy, dö,, d (S a 5) | which two expresssions represent the “extension” in the space of 6 dimen- sions after the collision. That before the collision is f ds dy d$ dg, dn, d5,, so that, when the determinant — 1, the extension remains un- changed by the collision. This proves really to be the case, as JEANS shows. We may, however also consider this property as a special case of the theorem of LiouvitLE, and derive it from this *). This theorem says, that with an ensemble of identical, mutually inde- pendent, mechanic systems, to which Hamiiron’s equations of motion apply, [erent = | dP,...dQn, When p,….gn represent the coordinates and momenta of the systems at an arbitrary point of time, P,... Qn those at the beginning. Gras calls this law: the principle of conser- vation of extension-in-phase, which extension we must now think extended over a space of 2n dimensions. When now the two collid- ing molecules are considered as a system which does not experience any influence of other systems, and it is assumed that during the 1) Cf. Lorentz, l.c. Abhandlung VII. *) As Boitzmann cursorily remarks: volume II p. 225, ( 498 ) collisions forces act which only depend on the place of the particles and not on the velocities, we may apply the formula f ap, ver dn en fr. ..-dg, to an ensemble of such pairs of molecules, the former e representing the extension-in-phase after, the latter that before the collision. In the case discussed by BorrzmanN the masses of these molecules are m and 7, so that we get: fee dy dz' m° d8 dy! dS’ de’, dy', de’, m,* dé’, dy’, a, = = fax dy dz m? dg dy d§ de! dy' de’ m,* dE dy dv. As we may consider the coordinates during the collision as inva- riable, it follows from this that: as dy dj ds. du, de, = fa dy ds ds, dy, ds. $ 3. However as has been referred to above, we may, without assuming anything about the mechanism of the collision, prove the property by means of the formulae for the final velocities with elastic collision, making use of the functional determinant. Another method is followed by Wiunp in his above-mentioned paper (the second proof) and by BOLTZMANN (vol. II p. 225 and 226); this method differs in so far from the preceding one, that the changing of the variables takes place by parts (by means of the components of velocity of the centre of gravity), which simplifies the calcu- lation'). A third more geometrical method is given by Wiuyp in his first proof. This last method seems best adapied to me to convey an idea of the significance of the principle of conservation of exten- sion-in-phase in this special case. | shall, however, make free to apply a modification which seems an abridgment to me, by also making use of the functional determinant. So it might now also be called a somewhat modified first method. In the first place I will call attention to the fact that with these phenomena of collision it is necessary to compare infinitely small volumes; if we, therefore, want to use the formula: a EL fi de! dy! dg d&', dy, a3, = | Re dg dy dö ds, dn, a EE 1) It seems to me that in this proof Botrzmann does not abide by what he himself has observed before (§ 27 and § 28, vol. I), viz. that the equality of the differential products means that they may be substituted for each other in integrals. The beginning of § 77 and the assumption of du dv dw, and dUdV dW, as reciprocal elements of volume, is, in my opinion, inconsistent with this ( 499 ) we must take infinitesimals of the 2" order. We can, however, also proceed in a somewhat different way. For how is the above formula derived? By making use of the fact, that with a volume d§ dy ds di, dy, d5, in the region of the §.. &, corresponds a volume ve (lagen | ae EN ds dy dS ds, dn, dö, =) 1 in the region of the &...&,, or also that the first mentioned exten- sion, occupied by the representing points in the space of 6 dimensions before the collision, will give rise to the second extension after the collision. We can, therefore very well compare these expressions inter se, without integration, if only the second expression is not interchanged with d5' dij d5' d3', dy, ds',, i. e. the volume element obtained by dividing the extension after the collision in another way. We now suppose the points of velocity before the collision to be situated in two cylindres, the axes of which are parallel to the normal of collision. The bases of the cylindres are dOdO, and the heights dd and dd,. The extension occupied by the combinations of the points of velocity is evidently equal to the product of the con- tents of the eylindres : dOdO, dddd,. In case of collision the compo- nents of the velocities perpendicular to the normal remain unchanged, so the points of velocity are shifted in the cylindres in the direction of the axis, so that d becomes J’, and d, becomes d',. Between ti : , md+ m, (2d these quantities exist the relations: d eres 9 m + m, pees A= 05). when m and m, denote the masses of the colliding m + m, molecules (i.e. the same relations as between the normal initial and final velocities with elastic collision. If we now wish to calculate the extension after impact we may make use of the fact that dO and dO, have not changed, so that we need only examine what happens to dddd, or what extension in the region of the d'd’, corresponds to the extension dddd, in the region of the dd, and d', (s'',) dddd , san ‚… and as it follows from the formulae for d' and d', that the absolute value of the determinant —1, the extensions before and after impact are equal. The extension after the collision is, however, not equal to the product of the cylindres in which the points of velocity will be found after the collision. This will be easily seen with the aid of the geometrical representation given by Wino. The extension before impact may be thought as the product of the extension in the space According to the above this is: ( 500 ) of four dimensions dOdQ, and the extension dddd,, which we may imagine as a rectangle in the region of the dd,, when we project them as two mutually normal coordinates in a plane. Every point in the rectangle represents therefore a number of combinations of velocities with equal d and d,. The sides of the rectangle with equations d= c and d, = c,, correspond in the region of the dd’, with the right lines md’ + m, (2d, — d) = (m + m,)c¢ and m,d', + m (2d — dd) = (m+ m,) ¢,, so that from the combina- tions represented by points within the rectangle after the impact others follow represented by points within an oblique parallelogram. 'd(d'd',), The formula abel | = 1 expresses that the two figures have the d(dd,) | same area. Now the extension after impact is equal to this paralel- N 07 a dd (501 ) logram XX dOdO, or the product of the two evlindres in which points of velocity were found before impact. The product of the cylindres, in which points of velocity are found after impact is equal to the product of dOdO, and the area of the rectangle with sides parallel to the axes O'd’ and O'd', described round the parallellogram under investigation. In this rectangle lie a number of points which have no corresponding points in the first rectangle. Only when m= m, rectangle and paralellogram coincide. Collisions of opposite kind, now, are such for which the combina- tions of velocity before impact are represented by points of the paralellogram in the plane d'Od', and after impact by points of the rectangle in the plane dd. Physics. — “Contributions to the knowledge of the w-surface of VAN DER Waars. XII. On the gas phase sinking in the liquid phase for binary mixtures.” By Prof. H. KAMERLINGH ONNEs and Dr. W. H. Kerrsom. Communication N°. 96% from the Physical Laboratory at Leiden. § 1. Zntroductvon. In what follows we have examined the equi- librium of the gas phase with the liquid phase for binary systems, with which the sinking of the gas phase in the liquid phase may occur. It lies to hand to treat this problem by the aid of w (free energy)-surfaces for the unity of mass of the mixture (VAN DER W aars, Continuität I] p. 27) for different temperatures construed on the coordinates v (volume of the unity of mass of the mixture) and & (quantity of mass of the second component contained in the unity of mass of the mixture). As VAN DER Waats (loc. cit.) has already observed, the laws refer- fing to the stability and the coexistence of the phases are the same for these w-surfaces as for the more generally used y-surfaces for the molecular quantity: in particular also the coexisting phases are indicated by the points of contact of the y-surface with a plane which rolls with double contact over the plait in the y-surface. In what follows we have chiefly to consider the projections of the con- nodal curve and of the connodal tangent-chords on the zv-plane. More particular cases as the occurrence of minimum or maximum critical temperature or minimum or maximum pressure of coexistence we shall leave out of account: we shall further confine ourselves to the case that retrograde condensation of the first kind occurs. Moreover we shall restrict ourselves to temperatures, at which the ( 502 ) appearance of the longitudinal plait does not cause any irregularity *). The component with the higher critical temperature (7%) is chosen as first component; its critical temperature is, accordingly, denoted by 7. The special case that 7;,=0, is that of a gas without cohesion with molecules having a certain extension. The investigation of the y-surfaces becomes simpler for this case. For the present it seems probable to us that helium still possesses some degree of cohesion. We will, however, in a following communication compare the case of a gas without cohesion with what the observations yield concerning mixtures with He. § 2. Barotropic pressure and barotropic concentration. We shall call v and w of the gas phase v, and z,, of the liquid phase v1, and a. At a temperature 7’ a little below 7}, we shall always have vv. For then the plait extends only little on the w-surface (see fig. 1), the plaitpoint is near the top of the connodal curve, which is turned to 21, and all the projections of the connodal tangent- chords deviate little in direction from the v-axis, the angle with 7 En increases regularly if we go from « = 0 pa along the connodal curve to the plaitpoint, but it has but a small value, when 7 — 7 is small. Only when we take for 7’ a value a certain amount lower than 7’, the plait extends sufficiently on the w-surface 7 to allow that v, =v, and A= 5. dd the v-axis, 4 p — are tg If at a suitable temperature 7’ we have substances as mentioned at the beginning, as e.g. helium and hydrogen at the boiling-point of hydrogen, we shall find the projection of a connodal tangent- chord denoting the equilibrium considered in the zv projection of the eas-liquid-plait on the y-surface for 7’; to reach it we shall have to ascend from wv == 0 along the connodal curve up to a certain value of the pressure of coexistence p, before 4, which itself is zero for ww 5 xt=0, can become 5 A pressure of coexistence p= py, under which wv, =v at the temperature 7’, we call a barotropic pressure for that temperature, the corresponding concentrations of liquid and gas phase the barotropic concentration of the liquid and of the gas phase at that pressure and that temperature. For when v,—v, with increasing pressure of coexistence p passes through zero at p= jp», we find in equilibria with pressures of coexistence above and below the value ( 503 ) In order to examine how a barotropic tangent-chord first makes its appearance on the plait on decrease of 7, we point out that with ; A .., dÔ extension of the plait from 7%, at first 7, remains positive all over Lv the liquid branch of the connodal curve, so that at first we have to look for the greatest value of 6 at the plaitpoint, where we shall denote its value by 6,7. When, however, on decrease of 7’ the plait extends over the w- surface, this need not continue to be the case, and we may find dé ; ae alternately positive and negative. This is immediately seen when we notice that this must always be the case when the plait extends all over the p-surface. If with decrease of 7’ the maximum value of 6 more and more in- creases, and 7’ has fallen so low, that the maximum of 6 somewhere x in the plait has just ascended to Bp then at this 7’ the condition for the barotropic equilibrium wv, =v, will be satisfied just for the corre- sponding tangent-chord, and only for this tangent-chord. The higher barotropic limiting temperature is then reached. On further decrease of temperature the barotropic tangent-chord will then split into two parallel barotropic tangent-chords, the higher and the /ower tangent- chord, which at first continue to diverge with further falling tempe- rature, so that the higher barotropic tangent-chord may even vanish from the plait through a barotropic plaitpoint, and then, at a lower temperature, make its appearance again through a barotropic plaitpoint’). At still lower temperature it follows from the broadening of the plait in the direction of the v-axis, which at sufficiently low tempe- rature renders the occurrence of a barotropic tangent-chord impossible, that the maximum of 9 falls again, and the barotropic tangent-chords draw again nearer to each other. At Dee the tangent-chords coincide again, and the lower barotropic limiting temperature is reached. At lower temperatures vj =v is no longer to be realized, and v, is always > w. Figs. 2, 3 and 4 represent different cases schematically. In the spacial diagram of the y-surfaces for different temperatures the barotropic tangent-chords supplemented with the portions of the con- 1) The latter supposes that 77/1, is not very great; in accepting the contrary we would come in conflict with the supposition that the longitudinal plait dues not become of influence. Moreover we preliminarily leave out of account the case that both barotropic tangent-chords follow one anotl er in disappearing or appearing through a barotropic plaitpoint. [Added in the translation]. ( 504 ) nodal curves between the lower and the higher tangent-chords form together a closed surface, which bounds the barotropic region. If “on. the: other hand Omar pl remains, till it has reached or TU . . . JE exceeded the value 3? and if not a second maximum value for 6 > . occurs on the plait, a barotropic plaitpoint will occur at the higher barotropic limiting temperature, whereas at lower temperature a single barotropic tangent-chord on the plait indicates the equilibrium with vj =. With decreasing temperature this barotropic tangent- chord will at first move along the plait starting from the plaitpoint, but at lower temperatures it will return, and finally (the occurrence of a longitudinal plait being left out of consideration) it will disappear from the plait through a barotropic plaitpoint at the lower barotropic limiting temperature. In this case the barotropic region is bounded on the side towards which the plait extends by barotropic tangent- chords, on the other side by the portions of the connodal curve which are cut off. It follows from the above that — when the occurrence of barotropic tangent-chords on the y-surfaces for a definite pair of substances is attended by the occurrence of barotropic plaitpoints — if 7, > Typis (higher baratropie plaitpoint temperature) or Ty << Tipu lower barotropic plaitpoint temperature there always exists at the same time a higher and a lower barotropic tangent-chord; if 7,7; > Ti > Top there exists only one barotropic tangent-chord. The nature of the barotropic phenomenon for He and H, may serve for arriving at an estimation of the critical temperature of He. According to the investigation of one of us (kK. See Comm. N°. 96 c.) it is probable that the appearance of a single barotropic tangent- chord for He—H, at the temperature of boiling hydrogen would point to Tye << about 2°, whereas on the other hand when Tire is higher, a higher and a lower barotropic tangent-chord is to be expected. Further that, as was already observed in Comm. No. 96 a. (Nov. °06) a barotropic tangent-chord can only appear in the gas- liquid-plait when very unusual relations are satisfied between the properties of the mixed substances, which for the present will most likely only be observed for He and H,. - Whether it is possible that more than one barotropic region occurs, and whether one or more barotropic tangent-chords can move from the plaitpoint past the critical point of contact, is still to be examined. Also whether it is possible that the lower barotropic limiting tem- perature descends lower than 7%,, so that fig. 5 might be realised. With regard to these questions too it is only of practical importance to ( 505 ) know in how far the properties of He and H, create that possibility. § 3. Barotropic phenomena at the compression of a mixture of definite concentration. What will take place in this case is easy to be derived from the foregoing survey of the different equilibria which are possible at a same temperature. For the further discussion we have to trace the isomignie line, the line of equal concentration (we = const.) for this mixture, and to examine the section with the connodal curve, the suecessive chords, and finally again with the connodal curve. In the description of the barotropic phenomena we shall confine ourselves to the more complicated case, that at the 7’ considered both a higher and a lower barotropic tangent-chord occur, after which it will be easy to survey the phenomena when only one barotropic tangent-chord appears. To distinguish the different cases we must divide tbe liquid branch of the connodal curve at 7 into an infra-(¢@=0 to x= xp;7, lower barotropic concentration of the liquid phase at T), inter- (eu;r to eusr) and supra-(@ = psp to «= x, )-barotropte part, and the gasbranch into the three corresponding pieces falling within and on either side of the region between the two barotropic tangent-chords (the lower bir and the higher 5,7) at that temperature. Whether the phenomena of retrograde condensation attend those of the barotropic change of phase or not depends on this: whether both barotropic concentrations of the gas phases fall below the plait- point concentration or not. Let us restrict ourselves in this description to the case that this complication does not present itself or let us only consider mixtures for which «< #,;. On compression the first liquid accumulates in the lower part of the tube for « << #43;7 and for zp > «# > UybsT, and in the higher part for as7 > > agr. On further compression, when VgbiT > tisT, Change of phase will take place once for mixtures of the concentration z, so that vossr > 2 > aguiq OF Bier > 2 > aur; it will take place twice for mixtures of the concentration x, so that zip > © > vysr. So the last remains of the gasphase will vanish above for «<< epir and for a) > 2 > xps7, and below for LisT > > tyr. If it is possible that over a certain range of temperature the barotropic tangent-chords get so far apart that “LlbsT > UgiT, Change of phase will again take place once for these tem- peratures for mixtures of the concentration «, so that ass7 > @ > wijs OF LT > > UUT. This description will, of course, only be applicable to He and H, when the suppositions mentioned prove to be satisfied, ( 506 j $4. Disturbances by capillary action. As is always tacitly assumed in the application of the w-surface when the reverse is not expressly stated, the curvature of the surfaces of separation of the phases is put zero in the foregoing discussion. If the curvature may not be ne- elected, e.g. at the compression of a mixture in a narrow tube, then, when the barotropic pressure is exceeded, the phase which has thus become heavier, will only sink through the lighter phase under it, when the equilibrium has become labile taking the capillary energy of the surface of separation into account. For this it is required that 6, has become larger than 5 to an amount of A6, which will depend on the capillary energy of the surface of separation and the diameter of the tube in which the experiment is made. Thus capillarity causes a retardation of the appearance of the barotropic phenomenon: both with increase and with decrease of pressure the barotropic tangent- chord must be exceeded by increase or decrease of pressure to a certain amount, before the two phases interchange positions. In this way the difference of pressure mentioned in Comm. N°. 96", (Nov. 1906, p. 460) between the sinking of the gas phase chiefly consisting of helium and its rising again at expansion (49 and 32 atms.) is e.g. to be explained by the aid of the following suppositions which are admissible for a first estimation. 1. that at — 253° and 32, resp. 49 atms. He is in corresponding state with H, at 150° and 160, resp. 245 atm., in agreement with the assumptions M ne 0. He = 7 Maven according to the ratio of the molecular refractive powers, Tim, —=1°.5 (according to O1szewski < 1.7); if the gas phase consisted only of He (molec. weight 4), the density at the temperature and pressures mentioned would be 0.062, resp. 0.081, and if moreover the liquid phase had the same density with the two pressures, 46,,, would have to correspond to a difference of density of + 0.01; owing to the fact that the two last mentioned suppositions are not satisfied, the difference of density will be smaller ; 2. that the capillary energy of the surface of separation between the phases coexisting at the above temperature and pressures is not many times smaller (or greater) than that of liquid hydrogen at that temperature in equilibrium with its saturated vapour, and that the latter may be derived from that of nitrogen *) by the aid of the principle of corresponding states. The gas bubble will then in a tube like that in which the experiment described in Comm. N°. 96a was made (int. diam. 8 mm.) only sink through the liquid or rise again, when 1) Bary and Donnan, Trans. Chem. Soc. 81 (1902) p. 907, “XT ‘JOA ‘“Wepsajsury ‘peoy [eAoy sSuipaooold A ost ra eel € "3u lan ‚„somngxru Axeurq roy oseyd prbm oy} ur Suryurs oseydses oy} UO IIX ‘SIEEM Jop ueA Jo ooujins-? 04} JO oSpojmoug oy} 0} UOIJNYLIJUOD, 'WOSTAN 'H 'M “Ad PUE SANNO HONITUANVI “H Jd ( 507 ) the difference of the radii of curvature of the tops of the bounding menisci exceeds that between 3 and 5 mm. ; a At those temperatures for which @nac < AED A Gap, the phenome- non of the phase which is uppermost at low pressure, sinking and rising again does not make its appearance in consequence of gravity alone. If this eondition is satisfied for mixtures of a definite pair of substances for every temperature between the lower and the higher barotropic limiting temperature, the phenomenon could only be realised for these mixtures by the aid of a suitable stirrer. § 5. Remarks on further experiments with helium and hydrogen. a. In the experiments mentioned in Comm. N°. 967 the gas phase proved to remain below on compression to the highest pressure which the apparatus will allow. When we repeat these experiments at a higher temperature (which may e. g. be obtained by boiling the hydrogen of the bath under higher pressure *)) it is to be expected that the barotropic pressure will first rise, as in the beginning starting from — 253° the gas phase will continue to expand more strongly than the liquid phase. At higher temperature the liquid phase begins to expand more strongly than the gas phase, but the mutual solu- bility plays already such an important role then that a definite expectation cannot be expressed, unless this, that on account of the retreating of the plait and the impossibility of the barotropic tangent-chord to reach the side of the hydrogen, the higher barotropic limiting tem- perature may be pretty soon reached. Also in connection with the estimation, which may be made from this concerning Tiz, it will be of importance to investigate whether with a suitable concentration and at a suitable higher temperature we may observe the liquid phase sinking after wt had first risen. That the phenomena at higher temperature, if the glass tube used should prove strong enough to bear the pres- sure, should be prevented by capillary action, is not probable, as capillarity together with the differences of density decreases at higher temperature; moreover in spite of capillarity the phenomena might be realised by the aid of a suitable stirrer. b. With decrease of temperature the limit is soon reached at which we meet with the solid phase. The question rises whether then the phenomenon: the solid phase, (the solid hydrogen) floating on the gas phase (chiefly the as yet still gaslike helium), might not be realised. “1 Or by using the vapour from boiling hydrogen in a separate vessel [added in the translation). ( 508 ) Physics. — “Contributions to the knowledge of the w-surface of VAN DER Waats. XIII. On the conditions for the sinking and again rising of the gas phase in the liquid phase for binary mietures,”” by Dr. W.H. Kerrsom. Communication N°. 96e from the Physical Laboratory at Leiden. (Communicated by Prof. H. KAMERLINGH ONNES). $ 1. /ntroduction. As has been observed in Communication N°. 96% (See the preceding paper) (cf. Comm. N°. 96a, Nov. ’06, p. 459, note 1) it lies to hand, to take as point of issue the y-surface for the unity of mass of the mixture considered, in the investigation according to vaN DER Waars’ theory of binary mixtures, of the sinking and subsequent rising of the gas phase in the liquid phase, i.e. the barotropic phenomenon. Two coexisting phases of equal density are joined on this y-surface by a tangent-chord whose projection on the w,v-plane’) is parallel to the a-axis. It has already been observed in Comm. N°. 965, that with decrease of temperature starting from the critical temperature of the first (least volatile) component such a barotropic tangent-chord may make its appearance in two ways: a. by the angle of inclination of the tangent to the plait in the jlaitpoint, 4,7, reaching the value of — at a certain temperature 7 pl / bol A and by its exceeding this value at lower temperature. b. by 4 showing a maximum and a minimum on the plait at a certain temperature, and by this maximum reaching or exceeding T . . . . the value of 5. Also in this latter case one of the two barotropic tangents-chords which then appear, might reach the plaitpoint at ” lower temperature, and thus become 6,; = —. a . . Tv . . In both cases in which O1 => at a certain temperature it should = be expected apart from complications as e.g. a longitudinal plait ete. (ef. Comm. N°. 964, p. 502), the deseription of which will be : f LA : given later on, that 4); becomes again =~ for mixtures of the same = ‘substances at a lower temperature. In the first part of this paper the conditions are discussed on which JT a plaitpoint with Ot = a barotropic plaitpoint, occurs on the tp-sur- 1) Cf. for the meaning of © and v Comm. N°. 960. LEERT REN ( 509 ) face, whereas in the second part the conditions are treated for the appearance of a barotropic tangent-chord on the plait. A. On the conditions for the occurrence of a barotropic-plaitpoint. $ 2. In a barotropic plaitpoint the isobar, which in a plaitpoint always touches the plait, must run parallel to the zeaxis. This gives the condition: Oy X en nr ae or Val ict hel) Moreover a section v =v of the y-surface with the limiting position of the tangent-chord must then have a contact of the 3rd order. This furnishes: Op \ OMEN 5 a =f) ; = ik | ots . (1 band c) The two conditions (1 5) and (1c) follow also by applying (J a) to the general relations for the plaitpoint of Comm. N°. 75 (Dec. 1901) p. 294. The same may be obtained from the property of the barotropic ; dw points on the connodal curve that there 5 along the connodal wv shows a maximum or minimum’), so that the substitution-curve Ow ak: mn const. (see for the substitution curves on the y-surface for the v molecular quantity Comm. N°. 597), touches the connodal line in these points. § 3. For a first investigation we shall use vaN DER WAALS’ equation of state: Tit sea P En eens al 2 v—b, Vv? (5) with an a, and 6, not depending on v and 7’ for a definite w. In this: 1) This property for coexistence with 7,7, is analogous to the property that ( = — p along the connodal line is a minimum or a maximum for coexisting dv Ow tric line v =v, which joins two barotropic phases on the J-surface, is equal to Ow phases with #3 —= xj. In the same way the mean value of (=) along an isome- 0 the value of ; Ow a) for these phases. 34 Proceedings Royal Acad. Amsterdam. Vol. VIII. ( 510 ) RR (USE ee se” done BERS ai ve” (Sa) if R, = R/M,, R, = R/M,, R representing the molecular gas-constant, M, and M, representing the molecular weights; we put for this first investigation a,, = Va,,4,,, so that a; == Wa (Er) + Ya, - a. ee and when we put for the molecular volumes 20\237 = bim + bees, the relation for br given by vaN DER Waars Contin. II p. 27, reduces to: by bi (Li) A Dr Br Gs ce tee We get then (van DER Waars Contin. II p: 28): wy = — RT In (v—b,z) — = + TIR, (la) la (le) + Rywline} . (A) § 4. Taking equations (3) into consideration, and putting 4 b l+< Pee: pape (5) we get by the conditions (1) eo 142 5 2 =| 2 db da Me Me en De 1—z dn 1 da PF oe ut ae + R, = rf = a —— Tr dx? . . . . (65) x oy 2} 1 cae ek 5 2) fe ce N a u+ —— }= ; Ly te Ga Ee) ee Oe Ve eae [es 9 These equations are sufficient to eenn the data for a barotropic plaitpoint wipt, Popi Top: for a definite pair of substances. Eliminating T from (6a) and (65), we get, taking (6c) into consideration and putting : (Wa. + Mall Ven Va) 2 2 u En aS (ute) +e je ‚® while elimination of v from this equation and (5), putting: bet Pr (6,,—5,,) =» SS Ee Sek ee ee (75) }(e—0) Ui yields: / 2 ej + | zutlj=0. (9) From this equation with (6e): 27,7 may be found for given R,/R,, wand», after which #7, vo,; and 7,7, as well as pp», follow easily. ‘| $ 5. That a barotropic plaitpoint exists on the liquid-gas-plait with the assumed suppositions (2), (84), (Sc) and with suitable Gea) values of the constants, appears as follows: for z=0.5, lng foe Ie ie) ee. yrelde:. «7%, == — 1.957, after which (9) yields: (© = — 1.176, so that a,,/a,, = 0.00653. Thus we find for a mixture of two substances with I/, = 2 M,,4,='/, Vp, (so that the ratio of the molecular critical volumes is '/,), 7%, = 0.052 ye ah barotropic plaitpoint for vj, = 0.26 vj, Tip: = 0.80 Ty, , pit = 4.8 py, (To be continued). ERRATUM. In the Proceedings of the meeting of September 29, 1906. p. 209, line 15 from the bottom: for § 10 read § 9. p. 210, Table I, line 5 and 4 from the bottom, for: 5 July read 6 July. EN from the bottom, for 3 March ’05 read 3 March ’06. (January 24, 1907). ¢ dovitig oe KA A jn * A SDA dt ears Ml Di x: : of KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN -TE AMSTERDAM, PROCEEDINGS OF THE MEETING of Saturday January 26, 1907. Sanaa ae IOC (Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige Afdeeling van Zaterdag 26 Janunri 1907, Dl. XV). GONE NES. A. F. HorreMAN and G. L. Vorrman: “a- and §-thjophenic acid”, p. 514. J. D. van Der Waars: “A remark on the theory of the #-surface for binary mixtures”, p. 524. W. A. Wisrnorr: “The rule of Neper in the four-dimensional space”. (Communicated by Prof. P. H. ScHoure), p. 529. P. H. Scnovre: “The locus of the cusps of a threefold infinite linear system of plane cubics with six basepoints”, p. 534. W.J. H. Morr: “An investigation of some ultra-red metallic spectra”. (Communicated by Prof. W. H. Jurius), p. 544. F. Scuun: “On the locus of the pairs of common points and the envelope of the common chords of the curves of three pencils”, (2nd Communication. Application to pencils of conics). (Communicated by Prof. P. H. Scroure), p. 548. F. Scuun: “The locus of the pairs of common points of four pencils of surfaces”. (Commu- nicated by Prof. P. H. Scmoure), p. 555, C. H. Winn, A.F. H. Datnuisen and W. E. Rincer: “Current-measurements at various depths in the North-Sea”, (!st Communication), p. 566. (With one plate). F. Scuun: “The locus of the pairs of commen points of n-+-1 pencils of (n—1) dimensional varieties in a space of n dimensions”. (Communicated by Prof. P. H. Scuoure), p. 573. H. G. van De Sanpk BAKHUYZEN: “On the astronomical refractions corresponding to a dis- tribution of the temperature in the atmosphere derived from balloon ascents’, p. 578. H. ZWAARDEMAKER: “An investigation on the quantitative relation between vagus stimulation and cardiac action, an account of an experimental investigation of Mr. P. Worrerson”, p. 590. Erratum, p. 598. 35 Proceedings Royal Acad. Amsterdam. Vol. IX. (514 ) Chemistry. — “a- and 8-thiophenic acid.” By Prof. A. F. HoLLEMAN and Dr. G. L. VOERMAN. (Communicated in the meeting of December 29, 1907). A very remarkable research on these acids was published in 1886 by V. Meyer, who discovered the same. In the main it amounts to this, that in addition to the theoretically possible monocarboxylic acids of thiophen | 2|CO,H | A|CO,H bvd 5 S Mp. 126°.2 Mp. 138°.4 a third isomer was obtained, called a-thiophenic acid m.p. 117.5—118°, which, however, in its derivatives such as the amide, the phenylurea derivative of the amide, the amidoxime and the thienone (C,H,5),CO, so completely resembled the corresponding derivatives of the a-acid that they would have been declared identical, were it not that from the a-derivates the a-acid m.p. 118° was regenerated, whilst the a-derivates yielded the «-acid m.p. 126°. V. Meyer expresses himself as follows: “Die Vergleichung der a- und «-Säure ergab immer von ‘“Neuem das merkwiirdige Resultat, dass die beiden Säuren wirklich ‘in ihren Eigenschaften durchaus verschieden sind, und dass die Ver- “schiedenheiten sich als constante, durch keinerlei Reinigung oder “Umwandlungen zu entfernende Eigenschaften erwiesen; dass aber “alle Derivate der beiden Säuren in ihren physikalischen Eigenschaften “absolut zusammenfallen und fiir identisch (im gewöhnliehen Sinne) «erklärt werden miissten, wenn sie nicht die Eigenschaft besässen, “dass jedes aus der a-Säure dargestellte Derivat bei der Rückführung “auch wieder «-Säure, jedes a-Derivat dagegen a-Säure lieferte.” As the a-acid had also been obtained by oxidation of tar-thiotolene which is a mixture of a- and 9-thiotolene [2—3 methylthiophen |, V. Meyer suspected that this a-acid might be after all a mixture of a- and g-acid, and he really succeeded, by oxidation of a mixture of the two thiotolenes in definite proportion, or by slow erystallisa- tion of a mixture of «- and g-acid from cold water, in obtaining an acid which agreed in every respect with the a-acid. This was no doubt an important step forward, but the behaviour of such a mixture and also of the derivatives obtained therefrom still remained a very remarkable one. 1) A. 236, 200; also V. Meyer, die Thiophengruppe p. 188—207. ( 515 ) Notwithstanding all this, nobody, since this elaborate research of V. Meyer, has been engaged during the last 20 years in the study of these acids, although it might have been expected from the deve- lopment of the phase-rule that the latter might possibly give us a closer insight into the phenomena described above. The probable cause of all this is that these acids are not readily accessible, and that those engaged in researches connected with the phase-rule have not ventured to prepare the same. When Dr. Vorrman, at my request, undertook the closer study of these acids we had, first of all, to find a better process for the preparation of these substances. In the ease of the a-acid we have indeed succeeded in a very satisfactory manner. We have also worked out another and improved method for the preparation of the g-acid; but it is still unsatisfactory owing to the small yield. Therefore, we have been obliged to restrict ourselves, provisionally, to the study of the acids themselves; the derivatives will be taken in hand when more material has been obtained. Preparation of a-thiophenic acid. By V. Merer and his pupils, this acid was best obtained by oxidation of propiothienon C,H,S.COC,H,, because the oxidation of the much more readily accessible acetothienon C,H,S.COCH, yielded a mixture of a-thiophenic acid and thienyl- glyoxylie acid, which it was rather troublesome to separate. We have succeeded in converting acetothienon almost quantitatively into a- thiophenie acid, being guided by the following considerations. If we oxidise a methyl ketone, experience has taught that the methyl group very readily changes to carboxyl thus forming a glyoxylic acid: R.CO.CH,—>R.CO.CO,H. If, however, we attempt to go further and obtain the corresponding carboxylic acid: R.CO.CO,H-R.CO,H a difficulty is experienced and the oxidising mixture then also attacks the group R so that the yield of the carboxylic acid becomes generally unsatisfactory. Now some time ago, I found a method for converting acids R.CO.CO,H quantitatively into R.CO,H; this is rendered possible by the application of hydrogen peroxide which causes a ready resolution according to the scheme: RCO CO. oe. | HO on” = R-COOH + CO, + H,0 This method has led to the desired result in this case. The oxidation of acetothienon is, therefore, done in two stages, first the formation of thienylglyoxylic acid which is subsequently oxidised to -thiophenie acid. The practical application of these processes was as follows: 35% ( 516 ) Acetothienon “was prepared from thiophen according to the method of Frreper and Crarts, and a very good yield was obtained. The thiophen was prepared by ourselves by distillation of sodium succinate with P,S;. 11.5 grams of ketone and 12 grams of sodium hydroxide were introduced into a litre of water, and to this was added slowly, at the ordinary temperature, a solution of 42 grams of potas- sium permanganate dissolved in a litre of water. After each addition the pink colour was allowed to change to green before addition of a fresh portion. After all the permanganate had been added the liquid was allowed to remain overnight; the solution was heated gently on the waterbath until the green colour had disappeared, then filtered off from the manganese dioxide, and concentrated to 250 cc. Without isolation of the thienylglyoxylic acid, beforehand the liquid, after being nearly neutralised with hydrochloric acid, is mixed with 9 grams of 30 °/, hy- drogen peroxide previously diluted with its own volume of water. The whole is set aside for a few hours, and afterwards heated for a few moments on a water- bath. On acidification the liquid the greater part of the z-thiophenic acid formed is precipitated in a pure condition; a further small quantity may be recovered from the mother-liquor by extraction with ether. By recrystallisation from water and distillation in vacuo, the acid may be obtained pure and quite free from thienyl- glyoxylic acid. The yield amounts to about 9 grams. The solution of a-thiopbenic acid saturated at 24°.9 contains 0.75 °/,. Preparation of B-thiophenic acid. V. Meyer has effected this by oxidising with potassium permanganate in very dilute, cold solution. The yield of g-acid was however very poor, in fact only about 5—8°/, of the thiotolene employed. After trying various modifications of this direct oxidation process without arriving at a better result we decided to follow an indirect way by first chlorinating the side chain, then preparing the aldehyde from the thienalchloride and finally oxidising the former to the acid: C,H,S .CH,— C,H,S . CHCl, — C,H,S .CHO — C,H,S . COOH. Bearing in mind van per Laay’s research’) on the bromination of toluene where it was shown that in presence of PCI, the sub- stitution in the side chain is accelerated, this substance was added in the chlorination of g-thiotolene. The above mentioned processes all proceeded very smoothly, but unfortunately an acid rich in chlorine was finally obtained as, apparently, the chlorination had also extended to the nucleus. This certainly could be freed from chlorine by treatment with sodium-amalgam but a large proportion of the g-thiophenie acid was lost thereby so that the yield did, finally, not exceed 10°/, of the thiotolene employed. We add a few particulars as to the modus operandi followed. G-thiotolene was prepared by ourselves by distilling sodium pyrotartrate with phosphorus trisulphide. The chlorination took place in direct sunlight in the presence of 10°/) of PCl. The reaction product is boiled in a reflux apparatus with water and calcium carbonate. The aldehyde is distilled in steam and purified over the 1) These Proc. Oct. 1905). (JMA) bisulphite compound. From 10 grams of thiotolene about 5 grams of the aldehyde are obtained. Of this, 3 grams are oxidised with 500 cc. of water containing 3.2 grams of potassium permanganate and 1.3 gram of 80°/) potassium hydroxide ; after standing over night the liquid is filtered from the manganese dioxide, con- centrated and acidified when about 3 grams of thiophenic acid are precipitated. The dechlorination of this product with sodium amalgam in dilute aqueous solution takes 15—20 days during which a large portion of the acid gets lost. As regards the solubility of 8-thiophenic acid at 25° it was found that the saturated solution contains 0.43 °/, of acid. Melting point line of mixtures of the two acids. According to the present views of the phase rule it was natural to suppose that the impossibility of separating these acids by crystallisation is due to the fact that they yield mixed erystals. In fact by determining the melting point line, Dr. Vorrman has succeeded in demonstrating with certainty that they give an interrupted mixing series. The initial solidifying points may be observed very sharply but the final solidifying points can only be determined within 0°.5 A list of the initial and final solidifying points is appended ; and in the annexed curve these figures are represented graphically. 740 740 * S++ a 130 © oa An 730° 125° 125° 120 120° 15° 115 110 110 105° 105 5 alo io Y3 0 10 20 80 40 50 60 70 80 90 100%3 off 4 atl ro LU 50 ife / sÛ 9 / ( 518 ) SOLIDIFYING POINTS OF « AND B-THIOPHENIC ACID AND THEIR MIXTURES. Uo a | Oo # | 1st solidifying point 2nd solidifying point | 100 0 126-9 99.01 0.99 195.4 123-9 194.4 98.25 1.75 195.4 | 96.56 3.44 124.6 193. 4193.6 94.3 5.7 124.4 121.0 93.60 6.40 193.4 90.60 9.40 422.2 88.20 11.80 121.3 + 118 85.82 14.28 120.3 + 116.6 85.0 15.0 120.1 79.45 20.55 447.7 143.5114 17.45 2955 117.9 412 —112.6 75.3 24.7 116.3 110.8—114.2 74.60 25.40 116.0 +110.6 69.45 30.55 114.3 40 —144 66.20 33.80 113.3 110.5—110.8 63.35 36.65 412.5 110.8 59.70 40.30 114.5 110.5 58.0 12.0 111.0 110.7 55.0 15.0 112.6 110.8 50.85 49.45 145.0 110.7 12.50 57.50 119.6 id 38.9 614 121.2 411.2 33.60 66.40 124.0 416.5—A17.5 23.80 76.20 498.2 123194 (+ 193.5) 14.0 86.0 132.6 199.5—120.8 5 94.6 136.3 134 134.3 0 100 138.4 It appears that the series of mixed crystals is interrupted on one side at 25°/, @-acid, and on the other side at 61°/,; and that there is a eutectic point at 42.5°/, B-acid at a temperature of 111°. (519 ) The erystallographical investigation of these acids and their mixtures kindly carried out by Dr. Jancer leads to exactly the same result. Dr. Jancer reports as follows : Of the two isomeric compounds B-Thiophenic acid erystallises the most readily in sharply defined, small crystalline plates. Whether obtained by crystallisation from solvents or by fusion and subsequent cooling, the compound exhibits the microscopical appearance of the subjoined figure. The crystals are monoclino-pris- 1A mjn IN, if. Fig. 2. Microscopcial aspect of «+ and 6-Thiophenic acid. matie, and combinations of the form: {001}, very predominant, {110} and {100}; the angle of inclination # deviates considerably from 90°, so that the smaller individuals often exhibit rhomboidal form owing to simultaneous development of {110} and {001}. Often the plates are so thin that only a single parallelogrammatic circumference can be observed with a very slight stunting of the sharp angle which was determined at 42°—43°, by {100}. In addition, small rectangular plates occur which, as the investi- gation shows, are formed witb {100} as predominant form, and therefore show prolongation along the b-axis. Although representing apparently a second form they are, however, quite identical with the parallelogrammatic phase. The optical axial plane is parallel {010} and falls along the longest diagonal of the parallelograms or perpendicular to the longitudinal direction of the needle-shaped individuals. In convergent light one hyperbole with rings is visible at the border of the field of vision. ( 520 ) Very feeble inclined dispersion with e >>v; double refraction nega- tive. The longitudinal axis of the parallelograms and the shortest dimension (breadth) of the more needle-shaped individuals are the directions of a smaller optical elasticity. a-Thiophenic acid crystallises from solvents or from the fused mass in long more or less broad needle-shaped individuals, which usually exhibit only a rudimentary limitation and cannot therefore be properly determined morphologically. Although the optical proper- ties seem to point to a monoclinic symmetry one might also feel inclined to conclude to a triclinic symmetry on account of the form limitations occurring here and there. The extinction of the needles is, however apparently orientated perpendicularly to their longitu- dinal direction. The smaller optical elasticity axis coincides with the longitudinal direction of the needles. The optical axial plane is orientated perpendicularly to the longitude of the needles ; in con- vergent light a single very characteristically coloured hyperbole is visible at about */, of the diameter of the fields of vision. Enormously strong dispersion with 9 < v; the sign of the double refraction around the correlated bissectrix* is positive. The two isomers are, therefore, readily distinguished microscopic- ally by the following properties : B-Thiophenic acid a-Thiophenie acid Parallellogrammic limitation, or Long, very slender needles, short rods of rectangular form. mostly with rudimentary limita- Very high interference colours. tion. One optical axis with elliptical Grey, or unconspicuous colours. rings, very weak dispersion @ < v. One coloured hyperbole ; very Negative double refraction. strong dispersion: @< v. Monoclinic symmetry ; angle of Positive double refraction. the parallelograms 42°—43°. | Triclinie or monoclinic sym- Optical axial plane for the paral- | metry. lelograms // to the longest dia- The optical axial plane is per- gonal for the needles perpendicular | pendicular to the longitudinal to the longitudinal direction. | direction of the needles. The latter The largest elasticity-axis is coincides with the shortest elas- parallel to the longitudinal direc- | ticity-axis of the erystals. tion of the needles or to the shortest diagonal of the paralle- lograms. ( 521 ) On account of the evaporation of the two substances when melting, one is obliged always to use a covering glass under the erystallisation- microscope. On mixing the two isomers I have noticed the following on melting and subsequently cooling the mixtures. a. Mixture containing 61,6°/, of B-acid yields exclusively mired crystals of the g-form; formation of minute traces of a-erystals is not improbable. 6. Mixture containing 42 °/, of g-aeid yields chiefly mcved crystals of the a-form; at the edges of the fused mass, however, are found also very small, slightly coloured parallelograms of the p-form. Negative mixed: crystals in the a-form (see d) were not noticed; only positive ones with 9 < v. c. Mixture containing 35,5°/, of g-acid behaves on solidification like 5. d. Mixture containing 22,5 °/, of B-acid, only yields mixed crystals of the a-form both positive and negative doubly-refracting but with o <v like the a-acid itself. e. Mixture containing 86°/, of a-acid only gives mixed crystals of the a-type with strong dispersion @ << v and a positive double refraction. Dr. JArerr comes to the following conclusion : “There exists here an isodimorphous mixing series with hiatus. This extends from a g-acid concentration > 22,5°/, to mixtures containing 61—62 °/, of the 8-compound. The mixed crystals of the a-type become on addition of the negative 8-compound less strongly positive optically and in the immediate vicinity of the hiatus even negative ; they, however still retain the strong dispersion with @ <v, which is so characteristic for the pure a-compound. On the other hand, the mixed crystals of the B-type have at all concentrations of 62—100°/, a negative double refraction and a very feeble inclined dispersion.” V. Meyer states in his treatise that in the oxidation of mixtures of the two thiotolenes, he has obtained various other mixtures of a- and g-thiophenie acid, and that these showed no sign of separation into their components when subjected to fractional crystallisation. Dr. Vorrman, however, cannot confirm this observation. When he recrystallised a mixture of 85.3°/, a-acid and 14.7°/, of g-acid (solidi- fying point 120°.3) from hot water, the solidifying point increased to 121 .6 which corresponds with a mixture of 89°/, a- and 11°/, of B-acid. As, however, V. Mryrer does not state the temperature at ( 522 ) which he carried out his fractional crystallisations, it is possible that this was not the same as in VorrRMAN’s experiment and this might account for the difference. Dr. VorrMAN has finally been engaged in the determination of the conduetivity power of the two acids and their mixtures in the hope of obtaining indications of a combination of the two acids when in solution. The observations are as follows : CONDUCTIVITY POWER OF a-THIOPHENIC ACID AT 25°, v | Ù | : 100 & 25 32.44 0.085 0.0314 50 45.37 0.118 0.0319 100 62.49 0.163 0.0319 200 85.06 0.222 0.0818 pag = 382.7 400 113.87 0.298 0.0315 800 1494 0.390 0.0312 (1600 189.34 0.495 0.0303] Average 0.0316 In this table, » represents the volume in which 1 mol. is dissolved, uw the molecular conductivity power, « the degree of dissociation, 100 & the dissociation constant according to OstwaLtp X 100. The conductivity power has been determined, previously, by OstwaLp (Ph. Ch. 8, 384) who found for 100% 0,0302, and by Baprr, (Ph. Ch. 6, 313) who found for 100% 0,0329. CONDUCTIVITY POWER OF g-THIOPHENIC ACID AT 25°. D) | p | 2 | 100 & 50 | 2835040) | 0.061 0.00783 100 39.82 |) 2 Odes vo) 80-0779 200 nee 0.417 | 0.007795 map = 382.7 400 62.06 0.162 | 0.00784 800 84.66 0.221 0.00785 [1600 114.17 0.298 0.00793] Average 0.00783 ( 523 ) CONDUCTIVITY POWER OF MIXTURES OF a + B-THIOPHENICG ACID AT 25°. 110/, thiophenic acid 89/9) 2 us Ih g. 100 & 50 43 .26 0.413 0.0288 100 59.49 0.155 0.0286 200 80.94 0.214 0.0284 400 108.05 0.282 0.0278 800 144 .29 0.369 0.0270 1600 180.75 0.572 0.026% 33.330/, 2 thiophenic acid 66 .660/, v Ii a 100 & So.000 31.79 0.083 0.0226 66.666 44 28 0.116 0.0227 193-933 60.86 07459 0.0226 266.666 82.33 0.215 0.0221 Boo Laoo 109.83 0.287 0.0217 1066. 666 143.38 0.375 0.0210 509/, Ê thiophenic acid 500/, « v be a 100 & 50 35.42 0.092 0.0185 400 48.52 0.127 0.0184 200 66.01 0.172 0.0180 400 88.58 0.231 0.0174 800 4AVTs45 0.306 0.0170 1600 451.7 0.396 0.0163 70.4%), 2 eigen acid 29.6"/, x p py a 100 & 100 AR OA | OAD” 0.0143 200 58.63 0.153 0.0139 400 | 79.06 0,207 0.0185 800 105.02 0.274 | 0.0130 1600 138.20 | 0.361 | 0.0128 ( 524 ) The conductivity of the pure d-acid has been determined previously by Loven (Ph. Ch. 19, 458) who found: - 200-4 = 00078: which agrees well with the value found by myself. The influence of the position of the sulphur atom in regard to the carboxyl group is very marked. From these observations it appears that the acids in aequeous solutions exert but very little influence on their mutual conductivity power, as the conductivity power of the mixtures agrees fairly well with the calculated result. A condensation of their molecules in such a solution cannot therefore be supposed to take place. Physics. — “A remark on the theory of the y-surface for binary mixtures” By Prof. J. D. van prR WAALS. (Communicated in the meeting of December 29, 1906). KAMERLINGH QONNks’ startling experiment, in which a gas was obtained that sinks in a liquid, bas drawn the attention more closely to the direction of the tangent in the plaitpoint of a binary mixture. Leaving the further particulars required for the realisation of sucha mixture to the investigations of KAMERLINGH ONNES and his collaborators, I will make a remark of general significance, in close connection with this experiment. In my Théorie moléculaire and more fully in Cont. II I have examined the condition, on which the tangent in the plaitpoint runs dv parallel to the v-axis, or in other words (5) =o. The problems at related to this may be reduced to 3. All three refer to the inter- 4 : dp : dp section of the two curves | — |= 0 and {| — |= 0. dv). de), As first problem I should like to regard the principal one, namely 2 2 that where — = 0 and — == 0. The point considered lies, there- Ov? dx Ov : 5 p dw fore, on the spinodal eurve, and at the same time the curve a 0 (0) 3 has for constant value of « two equal values for v and so also a =0; v Then the point considered is the critical point of the mixture taken as homogeneous. The value of 7’ is that of Tj. for such a mixture and the value of w is then found, when the approximate equation of state with 4 constant is applied, from : 1 da ade 2 de <a." b dx which value */, becomes =°/,, when the independence of 4 of the volume is relinquished. . “yy zage In this case the line cuts the line — = 0 still in two Or Ov Ov? points. One point is that above mentioned, the second lies at smaller v and larger v. So nearer the component with the smallest value of 5. With increase of temperature the two points of intersection draw nearer to each other, and as second problem we may put: to exa- mine the circumstances under which the two points of intersection of these curves coincide. The three equations from which this cir- cumstance is determined, are then : a mi on: Ov? dx Ov which expresses that these curves touch, viz. : dp we dw dy Ov? de) Oa? Ov dv? 0*p aes dp dp dxdv ) dc? dv? Above the temperature at which these circumstances are fulfilled, d*y Ory — —0O and =O do not intersect any longer, and the compli- dv? dxdv cation in the course of the isobars, viz. that there is one that intersects itself, has disappeared. The third problem is more or less isolated, but yet I should like 2 to treat it in this connection: viz. that for which the line In 5 0 1 dy O° yp ni 0 and PE OD 0. If there is a minimum 7) for mixtures taken us homogeneous, such a point is really a double point. If there should be a maximum 7%, it is an isolated point. We find then again v = vj, T'= 7; and the value of zw is that for which 7} has a minimum or maximum value. Let us call the three values of « obtained for those three problems wa, and #,; then: =0 and a third or has a double point, and so at the same time vs 0 vy = vs han” dij ey - ( 526 ) Now there are three more problems, and to this [ will call atten- tion in this note, which may be considered as the analogues to the three above-mentioned ones. If in the above problems we substitute the quantity « for » and 2 Daf 0? vice versa, so that — changes into — „and — — remains unchanged Ov Ow dwdv à 2 s 2 then the intersection of the curves a = 0 and are ae 0 will give av LUL rise to three problems, which are of as much importance for the theory of the binary mixtures as the three above-mentioned problems, Oy Op which relate to the intersection of —— = 0 and —— — 0. Ov? dwdv S : : Zp In the first place the points at which the two curves a —0 and U Ory = 0 intersect will belong to the spinodal curve, as appears Odo | 07 yp 0 dw \? en (Gn) In the second place these points of intersection will have the same MA dy significance for the course of the curves | — | = q = constant, as v Ow a wp 0? yp the points of intersection — = O and — Ov? Ord == 0 have for the course Ow of the curves (ze) — — p= econstant. The first point of inter- (hi r section will be a double-point for the q lines, whereas the other point of intersection will present itself as an isolated point, the centre of detached closed portions of the q lines. In the third place there will be a limiting temperature for the dw ) existence of the locus aS 0. With increasing value of 7’ this curve wv 20M) ) Ov? contracts to an isolated point, just as is the case with = 0 with Op maximum 7%, or as the curve Ie =—0 has a double point with Vv minimum 77. In the fourth place there is a temperature at which the curves 07 Ow — — (and — —0 only touch, and the two points of intersection Ow Owdv ; have, accordingly, coincided. And finally, and this is the most important case, there is a tem- (527 ) perature at which the intersection of these curves takes place in such a way that at one of the points of intersection a tangent may Op de be drawn to —-=0, for which = 0. Ox? da To determine these circumstances we have the three equations 0 dw = 0, _— = 0 and — =0 and this problem proves to be the Ow? ” dwdv Ou? | 5 . : 8 dp dw analogue of that mentioned above, for which — - = 0, — -= 0 and Ov Oudv dp dv da dv — == 0. If there — was =o, now —=o or — = 0. dv? da dv da By Wer 8 equations ee andy ite So if the 3 equations —— = 0, ——-=0 and —- == 0 admit ofa dw? 7” dade a solution, the circumstances may ke realised in which at the plaitpoint a tangent may be drawn // w-axis. Neglecting the variability of 4 with v we find for the three equations : een ee Op He MRT e de dx? ig = k x 2) 1 Oz? (le (v —b)® v () db? . MRT | — dw MRT — 2.) dx RA Fer ae Sy = EE it (2) Oz? ij EN) (v_—b)? db da ; MRT — On, da de : Oman) (2-5)? ae fink il pn) aaa If we puta = A + 2Bx + Ce’ and b= b, + a = b,+ 2 (b, — d,) we get the equation: we 2a" my ce r Ef 4 (B + Ca) (Cb,—BB) beed TB + Ca a C?xz B(1—2) | If B=a,,—a, should be small in comparison with a, ++-a,—2a,,)7= C2, b we get # equal to '/, by approximation, at least if 3 is also small. Then real values are found both for z and for 7 and v; only this value of 7’ can lie below the melting point in many cases, and consequently it cannot be observed. However, I shall not enter into a further discussion. I will only 2 ; LT point out, that for suitable values of 7’ the curve a0 represents U a closed curve, which contracts with increasing value of 7’, and may contract into a point. ( 528 ) . v . . . . In the problem, for which — =o at the plaitpoint, this case is ak ZR 6 dv. EN the transition for the cases where soe positive or negative. In the HY … dw nk same way in the problem for which — — 0 in the plaitpoint, this Ak 4 He dv ve ; 4 is a transition case between — positive or negative. So the cases du may also exist for which on the side of the small volumes, the Lee ee quantity — in the plaitpoint may have reversed sign. at wp op When we examine the shape of the curves - = 0 and — = 0, dvdv Ou? it appears that it is required for the realisation of the case, that when Wot fy da da ay aan ie positive, also ae and zn are positive, and that the calculated temperature must lie above 7% of the first component when we want to apply the result to the coexistence of gas and liquid phases. At the top we have the limiting case of two coexisting phases. If the tangent 1s // z-axis, the molecular volume is equal and the density will be proportional to mm, (1—«) + mr. Put lr wv 1 —a' ih — —/ Eeen ") 7 aa and em as ris and i qa os etd al v v 0) When (m,—m,) and (w’— x) have the same sign, d’—d is positive. As «’—wx is negative when the first component has the smallest size of molecule, m,—m, must also be negative, which is satisfied for helium and hydrogen. We can, in general, represent the limiting density of a substance by 5 and then the law would hold: When the most volatile substance has the greatest limiting density, the gas phase can be specifically heavier than the liquid phase. For Helium the limiting density is probably equal to that of the heavy metals. From the supposition that it is formed by splitting off from heavy metals this follows already with a certain degree of probability. 1) On further investigation it has appeared to me that a point that salisfies the dy OW Op naa — = 0, and = 0, possesses the analytical character of equations Dr — 0, Ande AEC p y a plaitpoint, but at least in many cases, does not behave practically as such. I hope to show this before long. (Added in the English translation). ( 529 ) Mathematics. — “The rule of Neper in the four dimensional space.” By Dr. W. A. Wryrrorr). (Communicated by Prof. P. H. SCHOUTE. (Communicated in the meeting of December 29, 1906). 1. The wellknown “rule of Neper” can in principle be formulated as follows: If we regard as elements of a spherical triangle A, A, A,, rectan- gular in A,, the two oblique angles A, and A, the hypothenuse a, and the complements of the two other sides } ar — a, and } z — a, ') we can apply to each formula generally holding for the rectangular spherical triangle the cyclic transformation (A, 4}u7—a,, a, 4$2—a,, A.) without its ceasing to hold. Fig. 1. We prove this rule by prolonging the sides A, A, and A, A, which (Fig. 1) for convenience’sake we shall imagine as < } 2, through the vertex A, = A’, with segments 4’, A’, and A’, 4’, so that A, 4’, = A, A',=4a. The spherical triangle A’, A’, A’, then proves to be again rectangular, namely in A’, whilst furthermore between the elements of both spherical triangles the following relations prove to exist: = ao Rr Bm kr a, Arde Anil Ax. From this is evident that the above mentioned cyclic transformation can be applied to the elements of each rectangular spherical triangle without their ceasing to be the elements of a possible rectangular 1) These are the complements of what Neper himself calls the “quinque circulares partes” of the rectangular spherical triangle. See N. L. W. A. GRAVELAAR, Joun Napier’s werken, Verh. K. A. v. W., First section, vol. VI, N°, 6, page 49. 36 Proceedings Royal Acad. Amsterdam. Vol, IX, ( 530 ) spherical triangle, from which further the rule of Neper immediately follows. The train of thoughts followed here will be found back entirely in the following. 2. A hyperspherical tetrahedron I shall call doublerectangular, if two opposite edges stand each normal to one of the faces. Let us suppose the letters A,, A,, A, and A, at the vertices of the tetahedron in such a manner that A, A, is perpendicular to the face A, A,A, and A, A, perpendicular to the face A, A, As. To make the tetrahedron doublerectangular it is necessary and sufficient for the angles of position on the edges A, A,, 4, A, and “AA. to be might”) a — = SN < a4 — Up SA — 3 Fs from which then ensues: / — | — — / —l As As = A == Ae — 7%, a he a == Gis a If we do not count the rectangular elements and if we count those which are equal only once the doublerectangular hyperspherical tetrahedron has 15 elements, namely a, @,3: 414: Aass Uzar Uzar G19) Css E34) Ars) Arn: EPE Ain: Ais Ass: 3. We now form, starting from a doublerectangular hyperspherical tetrahedron A,A,A,A, of which we think the edges all <t, a second hyperspherical tetrahedron (Fig. 2) by prolonging the edges meeting in A, — A’, through this vertex, namely the edge A, A, with a seg- ment A’, A',, the edge A, A, with A’, A’, and the edge A, A, with Al Al. so: that A, Ap == ALA = AL A a By very simple geometrical considerations we find that the tetra- hedron A’, A’, A’, A’, is again doublerectangular, that namely 4’, 4’, is perpendicular to A’, A, A’, and A’, A’, perpendicular to 4',4’, 4’; ; furthermore it is evident that the following relations exist between 1) The signs used here I have derived from Prof. Dr. P. H. Scnoure, Mehr- dimensionale Geometrie, Ist vol., page 267, Sammlung Scuusert XXXV, Leipzig, G. J. Géscuen, 1902. So | understand by dy the edge A, Aa; by «,) the angle of position formed by the faces lying opposite the vertices A, and Ag, i.e. the angle of position on the edge A; A,; by Aj the facial angle having A, as vertex and lying in the face opposite Ag, Le. the angle 43 A, A4. ( 531 ) Fig. 2. the elements of the two tetrahedra Je gg == Rr U tole a | a_ | & Es » ! . A= — di, 3 fh ae E: . ! — il Ay, = An ‚LL 6. ie Ai) Ay Ske ’ A. ‚Le. Ae zi (4 oO A,;) =32, A, <= Any 2 EE Me: (5 ha: Ais) eis (3 Mr Ai) Es ek a A, = 4% —4,, , | Ean Rn en ’ ! — Ay = A; 8 4. So if we regard instead of a,,, a,,, 4,,, A,, and a,, their complements as elements of the tetrahedron, then the elements of the 1) In giving the proof of this we must remember that A', A's and Ay Az lie on a sphere and therefore cut each other in a point P, just as A’, A'y and A, As cut each other in a point Q. 36% ( 532 ) doublerectangular hyperspherical tetrahedron can be arranged in three cycles, two of 6 and one of 3 elements, namely LG ee Bier BE a) ess Moes Baa); B (br Arin A, Gis nt Saas EA Aue 3. (A,,, 4% —a,,, A,,); 43 : so that it is possible to allow the elements of each cycle to undergo all simultaneously a cyclic transformation, if only afterwards those of the second cycle are replaced by their complements,') without these elements ceasing to be the elements of a possible doublerect- angular hyperspherical tetrahedron. These same simultaneous transformations may thus be applied to each formula holding in general for elements of the doublerectan- gular hyperspherical tetrahedron. 5. If we again apply the construction described in § 1 to the newly formed spherical triangle, etc. we find a closed range of five spherical triangles of which the hypothenusae form a spherical pentagon. The sides of these five spherical triangles are parts of five great circles on the sphere, namely the circles part of which is formed by the three sides of the original spherical triangle and the two polar circles of the vertices of its oblique angles. These five great circles form, however, another second similar range of five spherical triangles, namely that of the opposite triangles of the former range. 6. We can likewise deduce in a manner indicated in $ 3 out of a doublerectangular hyperspherical tetrahedron a range of such tetrahedra of which the faces all belong to six spheres, namely the spheres part of which is formed by the faces of the original tetra- hedron and the polar spheres of the points A, and A, Let us call B, the polar sphere of A,, B, that of A,, B, the sphere A, A, A;,, B, the sphere A, A, A,, B, the sphere A, A, A, and B, the sphere A, A, A Each of these spheres divides the hypersphere into two halves of which I shall designate the one to which the original tetrahedron 1) If we write the second cycle » (Gu Ay Ean eee oa Say a or (A A then no replacement of the elements by their complements is necessary, but the cycle has lost its symmetry with respect to the tetrahedron. pt Es Jl as a Daan oe Gis: Uzar 5 Aa oy eee A,,) ( 533 ) belongs by +, the other by —. The following list then indicates on which side of each of the six spheres the successive tetrahedra I, II, ete. are situated and by which they are limited. For the non- limiting spheres the sign has been placed in brackets. B, | Bg | B, | B, | B, Be dc rd ee ae ls RO ee |] - HIP |t |+ WI +])—-]+])0]MH}] - vi} —-} +} —-] - |W] 0 Wwe O}— | +] +] + io wl) o};o}-}-]- | - wi} =O |t |+ | + K) + )/—-)/O!lm|]—- | - x) —- | +f)—-|m}]o | + I} +) — | +) + [OO] a eet a le | | GE) Tha | db) + | + | + + It is clear that the tetrahedra I and VII are opposite to each other, likewise II and VIII, III and IX, ete., whilst the tetrahedron - I again follows tetrahedron XII. Thus the whole range consists of 12 tetrahedra which are two by two opposite to each other, in contrast to what we found in the three- dimensional space, where two ranges of spherical triangles are formed of which one contains the triangles opposite to those of the other. 7. Between the volumes of each pair of tetrahedra belonging to the range exists a simple relation. If we call Vz the volume of the first tetrahedron then the relation: dV] — $ dis dat, = $ ay, det; oe 4 4;, da, holds for each variation of the tetrahedron remaining doublerectan- gular (thus a, e,, and e,, not changing). Likewise dVj — 4 (Er Tk a,,) da,, = $ (47 pe ds.) da, ET $ (Er En @ts 4) da,, ¢ ( 534 ) So therefore Vi + I= rr + i aa,, — ha, (4a — a,,) + constant. The constant is found by putting a,, equal to a,, = 4, = 4,, = 3%, in which case V7 takes up the sixteenth part of the whole hyper- sphere, i.e. ta? whilst Vzz becomes = 0. The constant then proves to be —. 4 2, hence Vii ip —— ae Ct. LR Ep Fa, ($0 — 044): Likewise we find. Vu + Vin =- in +inlhr-a,) Hira, 4 (bra) (bar — dk Vu Viyv=-;a Hira, +124 (h0 —a,,) — kas, (4% —- @,,), ete. Every time the sum of the volumes of two successive tetrahedra can be expressed by means of four successive elements of the first cycle mentioned in § 4. We deduce easily from this: Ve Vi § My 503, — 34,, (5% — Cs): whilst in like manner we can find Vy — Viv, Vii— Vr, etc. Further we find V7 + Viv = ha,,a,, — Hag, (4% — a,,) — ba, (4 — ay,) and in like manner V7zz7 + Vy, ete. If we remember that the tetrahedra I and VII are alike with respect to their elements and volumes, II and VIII also, ete. and that with respect to the volumes we have to deal with only a closed range of six terms we see that of each arbitrary pair always either the sum or the difference of the volumes can be expressed in a simple manner. Mathematics. — “The locus of the cusps of a threefold infinite linear system of plane cubics with sie basepoints.’ By Prof. P. H. SCHOUTE. In the generally known representation of a cubic surface S* on a plane a to the plane sections of S* correspond the cubics through six points in a; here to the parabolic curve s'* of S* answers the locus C° of the cusps of the linear system of those cubics. The principal aim of this short. study is to deduce from wellknown properties of s'? properties of c'* and reversely. 1. If a plane rotates around a right line / of S* the points of intersection of that line / with the completing conic describe on / an involution, the double points of which are called the asymptotic points of 7. According to the condition of reality of these asymptotic ( 535) points the 27 right lines of S? supposed to be real, are to be divided into two groups, into a group of 12 lines with imaginary asymptotic points, the lines a, a | bb. B 2 3 4 5 of a doublesix and into a group of 15 lines c¢,,,c¢,,,...,¢,, with real asymptotic points. If to the six basepoints A’; of the linear system of the cubics the six lines a; correspond — and this case we shall in the following continually have in view — then to the six lines 6; correspond the six conics 6 through all the basepoints except A’; and to the fifteen lines cz, correspond the connecting lines c' —= (A';, A’,), whilst to the systems of conies (a;) in planes through a;, (b;) in planes through 6;, (ci) in planes through cz correspond successively the pencils of the curves of the linear system with A’; as doublepoint, the lines (6';) through A’; and the conics (ej) through the four basepoints differing from A; and A,. The situation of the six points A’; is then such that each of the fifteen lines c' is touched in real points by two conics of the pencils (cx), whilst on the other hand the points of contact of the tangents out of the points A’; to the conics 6; are imaginary . . . . . 2 . . so that each point A’; lies within the conic 6; with the same index. 2. As a matter of fact all real points of a line / of S* are hyper- bolic points of this surface with the exception of the two asymptotic points of this line showing a parabolic character; whilst each of these asymptotic points is point of contact of / with a conic lying on S*, / touches in both points the parabolic curve SS". If we apply this to each of the six lines a@;, imaged in the points A, and if we consider that to a definite point P of a; corresponds the point P lying infinitely close to A’; connected with A’; by a line of detinite direction (Versl., vol. I, pag. 143) we find immediately : “The six basepoints A; of the linear system are fourfold points of the curve c™ of a particular character, consisting of the combi- nation of two real cusps with conjugate imaginary cuspidal tangents, the cuspidal tangents of the curves out of the system with # cusp in A’;”’. The twelve points of intersection of the line ec’ with c'* consist of the isolated points A';, A’, counting four times and the real points of contact with two conics out of the pencil (cz) counting _ two times. Likewise do the 24 points of intersection of the conic b; with e'* consist of the five basepoints differing from A’; counting four times and the imaginary points of contact with the tangents through A‘; counting two times. 3. From the investigations of F. Kiem and H. G. Zrvruen dating from 1873 and 1875 it has become evident that the surface S* with 27 real right lines has ten openings and the parabolic curve s'* has ten oval branches. In connection with this we find: “The locus c'* has ten oval branches.” We ask which situation of the six basepoints A'; corresponds to the particular case of the “surface of diagonals” of CLEeBscn, in which the ten oval branches of the curve s'? have contracted to isolated points. In this case the fifteen lines with real asymptotie points, ie. in our case the lines cz, pass ten times three by three through a point; this is satisfied by the six points consisting of the five vertices of a regular pentagon and the centre of the circumscribed circle. What is more, each six points having the indicated situation can be brought by central projection to this more regular shape. The ten meeting-points of the triplets of lines then form the vertices of two regular pentagons (fig. 1). The curve c'* corresponding to these six basepoints then consists of merely isolated points, namely of fourfold ( 537 ) points in the six basepoints and twofold points in the ten meeting points of the triplets of lines. The remark that the curve c'? belonging to the six basepoints of fig. 1 has the line c’,, as axis of symmetry and transforms itself into itself when rotating 72° around A',, enables us to deduce in a simple way its equation with respect to a rectangular system of coor- dinates with A’, as origin and c’,, as x-axis. The forms which pass into themseives by the indicated rotation are e=24+ 77, P=2 — 10a*y’ + day*, Q= 5aty — 10a?y? + 45. If we pay attention to the axis of symmetry and to the identity P? + Q? =v" the indicated equation can be written in the form o* + ap’ + bo + co" + do" + P(e + fo? + go" + ho") + P*(i+h0%) = 0, so that we have to determine only the ten coefficients «,b,..,k. If now the common distance of the points A’,, A',,.., A’, to A’, is unity, then 3—e\’ 3 + eN? vt tat EE (wvl) | @ + 5 == 0; where e stands for 5, represents the twelve points of intersection of the curve with the z-axis. By performing the multiplication this passes into mt (a? + 2x" — Tat — Ov? + 202* — 62? — 72? +22 + 1)= 0. From this follows ee ga eae Aw Gee a dk =d — — 6, a oe Ct ne Oe ia SO, g So the equation go — 79° +209" — 79" +9" +2P (1 — 39? — 3049") — Q? (i +49") =0 is determined, with the exception of the coefficients 7 and & still unknown. Now the parallel displacement of the system of coordinates to A’, as origin furnishes a new equation, of which the form (Ai) y? 2 (12—4i—5d)ay? +a + (54 —281—45 hat? + (54-464 Bh)y! represents, after multiplication by 25, the terms of a lower order than five. The new origin being a fourfold point of c’? and the terms with 7° and ay? having thus to vanish, we find te A k= — 4, on account of which the indicated form passes into (a? + 5y?)?. The correctness of this result is evident from the following. Just as the two tangents in the old origin counting two times are represented by a + y? = 0,and therefore coincide with the tangents out of A’, to the conic through the other basepoints, so «* + 5y? = O represents ( 538 ) for the new origin A’, the pair of tangents out of A’, to the conic through the other basepoints. Or, if one likes, just as 2* + 7? is with the exception of a numerical factor, the fourth transformation (‘‘Ueberschiebung’’) of the first member of the equation (—O of the lines connecting A’, to the remaining basepoints, so #° + 5y° represents, likewise with the exception of a numerical factor, the fourth transformation of the first member of the equation fs = 0, which indicates with respect to the new origin A’, the five lines connecting A’, to the remaining base- points. . Finally the equation of c'* is e*(e°—7e° + 209*— 79? + 1) + 2P(9°— 39° —89* + 1)+ 4Q*(9*—2)=0, (1) or entirely in polar coordinates (9, g) 4(9?-2)9°cos 5p=(07 + 1)(o-49? + 1) = (97-1)*V (9*-L(49? + (59-1). (2) It is easy to show that this curve admits of no real points differing from the six basepoints A’; and tbe ten points of intersection of the triplets of connecting lines. If for brevity we write (2) in the form Leos5p=M + VN, then we tind — DT) sin? bg = (M7 + N=] 1) EAMYN wo oe and M? +N — L? = 2 (0?—1)? (29° —1) (0° —60°+-140*+-29’—1) (M?--N—L*)? — 4M? N = Ag" (o*—1)! (¢?--2)' (0'— 70? ++ N (4) If now we moreover notice that MN is negative and therefore _ 9 cos 5y complex when go? lies between — and 1, the following is Qt immediately evident : a. The first member of the second equation (4) tends to zero, when 0? assumes one of the values 0,1, 2, = (7 +-.3e); it is positive for all other values of 9’. b. If VN is real and 9? differs from unity the second member of the first equation (4) is positive; for the equation o® — 60° + 140% + 29? —1=0 : AE, 1 has, as is evident when the roots 9? are diminished by 1 De besides DE 1 one real negative root only one real positive one between rm and 1. 1 ce Af o° differs from’.0, 1, 2, =r! + 3e) the second member of ( 539 ) (3) is positive when MN is positive, and therefore p is imaginary. d. Neither does 9? — 2 give a real value for g; for substitution 3 in (1) furnishes for cos 5g the result zv? e. So we find only the real points : e= 0, p indefinite: . 3... A! ae Od ae i pe a. Aly Al AN Al Al ote 5 nn cossp=—l . . . the ten points of intersection of the fifteen connecting lines three by three. 4. We now consider a second case, in which the position of the six basepoints is likewise a very particular one, where namely these points form the vertices of a complete quadilateral. Through these six points not one genuine cubic with a cusp passes. For the three pairs of opposite vertices (A,, A,), (B, B), (C,, C,) of a complete quadilateral (fig. 2) form on each curve of order three, containing them, three pairs of conjugate points of the same system, and these do not oecur on the cubie with a cusp, because through each point of such a eurve only one tangent touching the curve elsewhere can be drawn. In this special case the locus of the cusps has broken ( 540 ) up into the four sides of the quadrilateral each of those lines counted three times. For it is clear that an arbitrary point of the line A,B,C, e.g., as a point of contact of this line with a conic passing through A,, B, C,, represents a cusp of the linear system of cubies. We can even expect that each of the four sides must be taken into account more than one time, because each of those points instead of being an ordinary cusp is a point, where two continuing branches touch each other. And finally the remark that the sides of the quadrilateral divide the plane into four triangles e with elliptic and three quadran- gles 4 with four hyperbolic points, so that they continue to form the separation between those two domains, forces us to bring them an odd number of times into account, namely three times because we must arrive at a compound curve c'*. Some more particulars with respect to the domains e and h. The nodal tangents of the cubic (fig. 3) passing through the three pairs of points (A,, A,), (B, B), (C,,C,) and having in P a node, are the double rays of the involution of the pairs of lines connecting oes Fig. 3. P with the three pairs of points mentioned, so also the tangents in P to the two conies of the tangential pencil with the sides of the quadrilateral as basetangents, passing through P; now, as these two conics are real or conjugate imaginary according to P lying in one ( 541 ) of the three quadrangles 4 or in one of the four triangles e, what was assumed follows immediately. To the case treated here of c’? broken up into four lines to be counted three times corresponds the parabolic curve of the surface S? with four nodes. 5. In the third place we consider stil the special case of six basepoints lying on a conic, in which the linear system of cubics contains a net of curves degenerating into a conic and a right line; in this net of degenerated curves the conic is ever and again the conic c’ through the six basepoints and the right line is an arbitrary right line of the plane. This case can in a simple way be connected with a surface S* with a node VO. If we project this surface out of this node O on a plane a not passing through this point, then the plane sections of the surface project as cubics passing through the six points of intersection of a with the lines of the surface passing through O; because these six lines lie on a quadratic cone, the six points of intersection with « lie on a conic. Besides, the sections with planes through O project as right lines; therefore the completing conic c? must evidently be regarded as the image of the node 0. Of course we must here again think that c? corresponds point for point to the points of OQ? lying at infinite short distance from O'; for c? is the section of @ with the cone of the tangents to S* in 0. As c? with one of its tangents represents a curve of the linear system, this conic belongs at least twice to the locus of the cusps. Here too this locus of cusps improper with continuing branches must be accounted for three times, so that the locus proper is a curve c° of order six, touching c? in the six basepoints. Let us suppose that c? is a circle and that the six basepoints on that circle (fig. 4) form the vertices of a regular hexagon, then the curve c° has the shape of a rosette with six leaves having the centre O' of the circle and the points at infinite distance of the diameters A,A,, A,A,, A,A, as isolated points. Of the ten ovals there are four contracted to points, whilst the six remaining ones have joined into the, circle of the basepoints and the curve c'. If we take point O' as origin and the line O'A,*as z-axis of a rectangular system of coordinates, then if O'4, is unity of length we find for the equation of c° dy? (y? — Ba)? +9 (a? HP} — 9 (a? + y?) = 0. It is evident from this equation that the curve c° can really stand Fig. 4. rotation of multiples of 60° round @', for then 2? + y? and y(y? — 32? are transformed into themselves. Out of the equation 3 sue Sip = ae 5,3 VI on polar coordinates it is evident that the curve c° (with the excep- tion of its four isolated points) is included between the circles de- 1 scribed out of O' with the radii 1 and So OE If we pass from the locus of the cusps to the parabolic curve of S* we must notice that the last curve has the node O of S* as threefold point, because c? has separated itself three times from the locus c’?. So this parabolic curve is an s° of order nine, a result which will presently be arrived at in an other way. We shall give — without wishing in the least to exhaust this case of the six basepoints situated on a conic — some degenerations of the remaining curve c° corresponding to some definite coincidences of the basepoints. a) The cases (2,2,2), (4,2), (6). If the six basepoints coincide two by two in three points of the conic, then c° consists of the sides of the triangle of the basepoints counted double, originating from compound cubics with a double line; there is not a locus proper. In reality the case (2, 2,2) of a conic touching in three points cannot occur for a genuine cubic with a cusp. | ( 543 ) The cases (4,2) and (6) are to be regarded as included in the preceding. By allowing two of the vertices or the three vertices of the triangle just considered to coincide we find for case (4, 2) the connecting line of the two basepoints counted four times and the tangent to the conic in the basepoint of highest multiplicity counted two times, for case (6) the tangent to the conic in the point counting for six basepoints counted six times. That there can be no locus proper in the last case ensues also from the fact, that a genuine cubic with cusp allows of no sextactic point. | b) The case (3,3). If the six basepoints coincide three by three in two points of the conic, then c° consists of a part improper, the connecting line of the two points counted four times, and a part proper, a conic touching the conic of the basepoints in these points. The new conic lies outside the conic of the basepoints. c) The case (1,5). This case agrees in many respects with the preceding. We find a part improper, the tangent in the point counting for five basepoints drawn to the conic of the basepoints, and a part proper, a conic touching the conic of the basepoints in these points. The new conic lies side the conic of the basepoints. 6. Of course it is possible to call forth by the curve c’? succes- sively all the different special cases which can put in an appearance by the parabolic curve s'* of the various surfaces S*. As this would lead us here too far, we limit ourselves to a single remark, which can eventually facilitate an analytic investigation of this idea. According to the general results with respect to a linear system of curves c” obtained as early as 1879 by HE. Caroraur the locus c42n—3) of the cusps of this system has in each 7-fold basepoint of the system a 4(2r—1) fold point and besides 6(2—1)’—2(37r?— 27r-+-1) nodes C. Each of those points C’ is characterized by the property that each curve of the system passing through this point is touched in this point by a definite line c. For the case under observation, „== 3 of the cubies, the number of points C is represented by 24—6p, when p is the number of basepoints. If we wish to investigate analytically what peculiarity the locus of the cusps shows in a basepoint of the system, or how a line through three basepoints separates from it, then the result — and this is the remark indicated — will be independent of the fact, whether the remaining basepoints occur or not, if in the former case, that some of these basepoints appear in a real or in an imaginary condition, we assume that these points both with respect to each ( 544 ) other and to the former basepoints have not a particular position. With the aid of this remark we can easily find the following theorems, with which we conclude: “Both cusps of which the fourfold point of the curve ¢,, coin- ciding with a basepoint A’; seems to consist and the two cusps of the curves of the system showing in this point a cusp, coincide in cuspidal tangents, but they turn their points to opposite sides.” “If the three basepoints A’,, A’,, A’, lie on a right line /, the locus proper of the cusps reduces itself to a curve c° touching the line / in A, A, A’. If the three remaining basepoints exist then the points of intersection of / with the sides of the triangle having those basepoints as vertices are points of c*”. The last case answers to that of a surface S* with a double point; the parabolic curve having in this doublepoint a threefold point, because / separates itself three times from c'*, is as has been found above already a twisted curve of order nine. Physics. — “An investigation of some ultra-red metallic spectra.” By W. J. H. Morr. (Communicated by Prof. W. H. Juztus). (Communicated in the meeting of December 29, 1906). Among the spectra of known elements those of the alkali-metals, by their relatively simple structure, lend themselves particularly well to an investigation of their ultra-red parts. Many observers have consequently sought for emission lines of these metals in this region. For the first part of the ultra-red spectrum the photographic plate may be sensitised; especially LEHMANN *) measured in this way various lines with wave-lengths ranging to almost 1 u. By means of the bolometer SNow ?) could advance to 1.5 u. For the further region, however, nothing was known about these spectra. CoBLENTZ*), to be sure, was led by a series of observations in this respect, to the conclusion that the alkali-metals emit no specifie radiation beyond 1.54, but I had reason to doubt the validity of this conclusion. In what follows I will briefly describe the method by which some ultra-red spectra were investigated, and the lines thus found. In an 1) H. LEHMANN. D.’s Ann. 5, 633, 1901. 2) B. W. Snow. W.’s Ann. 47, 208, 1892. 3) W. W. Coszenrz. Investigations of Infra-red Spectra. Carnegie Inst. Washing- ton. 1905. ( 545 ) academical thesis, which will soon be published, further details will be given. For the investigation of the alkalies, the metallic salts were volatilised in the are in the ordinary way. The very complicated band-spectrum, emitted by the are when no metallic vapour is present, extends far into the ultra-red. But this interferes in no way with the investi- gation of the metals, since it is entirely superseded when the are con- tains a sufficient quantity of metal. On the other hand the continuous spectrum, emitted by the incandescent particles in the arc, makes it somewhat difficult to observe some feebler lines; besides, the radiation of carbonic acid, the product of combustion of the carbons, (with a maximum near 4,44 u) persists with almost unchanged intensity. The image of the are is projected by a concave mirror on the slit of a reflecting-spectrometer ; the rays are analysed by a rock- salt prism and part of the so formed spectrum falls on a linear thermopile. This thermopile, like that of RuBens, is built up of iron and constantan; all the dimensions were chosen smaller than in the original pattern and a great sensitiveness was obtained. As well the emitting slit as the thermopile are mounted in fixed positions; in order to throw on this latter different parts of the spectrum in succession, the prism can be rotated through small angles. A WapbwortH combination of prism and plane mirror maintains minimum- deviation during rotation. In chosing and designing the instruments, the desirability,was kept in mind of replacing the very tiring reading of the galvanometer and the simultaneous noting of the corresponding position of the prism, by an automatical recording-device. [ had in mind the splendid arrangement by which Laneigy has for years recorded the intensity-curve of the ultra-red solar spectrum on a photographic plate. That this method has not been followed for recording heat- spectra instead of the time-absorbing visual observations, must be ascribed in the first place to a very complicated mechanism being required for obtaining complete correspondence between the linear displacement of the photographic plate and the rotation of the spectro- meter, and secondly to the difficulty of keeping the surrounding temperature perfectly equal during the observations. With very simple means I devised a method of recording, which avoids these two difficulties, while yet it warrants a sure “corres- pondence”’, and yields accurate results also when changes in the surrounding temperature cannot be prevented. For this purpose the continuous recording has been replaced by the marking of a series 37 Proceedings Royal Acad. Amsterdam. Vol. IX. ( 546 ) of dots, while for the continuous rotation of the spectrometer an intermittent one has been substituted. In this way for any recorded radiation-intensity the corresponding position of the prism can be found, not by measuring abscissae, but by counting dots. Since moreover not only the deflections of the galvanometer but each time also the zero- positions are recorded, it is possible to determine on the spectograms the radiation-intensities also when during the observations the surround- ing temperature, and consequently the zero-position, was variable. The principal advantages of this method of observation over the usual one are: 1. the absolute reliability of the observations, 2. the very short time required for a set of observations, 3. the accuracy with which interpolation is possible when the zero-position shifts, +. the non-existence of disturbances, caused by the proximity of the observer, 5. the complete comparability of the different observations, 6. the possibility of estimating the probable error from the shape of the zero-line. The short time in which a set of observations is made, is of importance when e.g. heat-sources are investigated which, like the are, show slow changes in radiation-intensity. A spectrum, ranging from 0,7 to 6u was recorded with 200 displacements of the spectro- meter in two hours. In the spectrograms a spectral line is represented by 5 to 6 dots. With one displacement of the spectrometer namely the line is shifted over a distance amounting to */, of the breadth of the image of the slit, or of the equal breadth of the thermopile. Hence the same kind of radiation will strike the thermopile during five successive displacements. From the mutual position of the dots, the place where the radiation-intensity has its maximum may be accurately determined. In order to derive from this the place occupied by the line in the spectrum, it is sufficient to know one fixed point in the spectrum. This fixed point was as a rule taken from a comparison spectrum, for which the carbonic acid emission of a Bunsen flame was chosen, the maximum of which, according to very accurate measurements of Pascuen, lies at 4.408. Part of the flame spectrum was for this purpose recorded simultaneously with the spectrum to be studied. A simple calculation then gives the refractive index for the unknown ray. In order to derive from this the wave-length of the line, a dispersion formula must be used. I became aware that the ( 547 ) well-known dispersion curves of Lanerey and of ReBeNs show considerable differences, and although at first sight Lanerey’s deter- minations seem to be much preferable, yet on closer examination _their excellence must be doubted, especially for the longer wave- lengths. To prefer one of the dispersion curves to the other seems to be at present a matter of arbitrary choice. So I have given in the tables besides the observed refractive indices, the wave- lengths, calculated from them as well by Lanenuy’s as by Ropes’ formula. The refractive indices hold good for a temperature of 20°; their determination is based on the index 1.54429 for the D-line, a value, derived from very accurate determinations by Lanexey. The tables given below contain the lines of Na, K, Rb and Cs (I have been unable to obtain reliable results with Li in the arc) and of Hg. The results were derived from a large number of, spec- trograms (10 to 12 for each metal). For the investigation of the mercury spectrum a mercury arc-lamp was devised, furnished with a rock-salt window. The spectrum of mercury has been repeatedly investigated as far as 10u; no measurable emission has been found beyond 1.7u. In the tables the first column gives the refractive index n of rock-salt, the second and third the wave-length u of the line, according to the formulae of LANerey and RuBeNs, and the fourth the approximate value / of the intensity. For the lines of which the exact position was difficult to ascer- tain, the refractive index is only given in four decimals. SODIUM. eas POTASSIUM. | 7 (Langley)! (Rubens), J na »(Langley)| (Rubens)! J | 1.53529 0.819 0.816 | 240 1.53654 0.771 0.768 | 620 53062 1.44 1.13 180 5325 0.97 0.96 10 „52961 Eb, 195 1531) ee 5810 1.44 1.10 20 5286 1.44 | 1.49 ae „53030 1.18 4.47 320 5281 1.57 1.54 5 | 52972 1.25 1.24 200 521 | 1.85 1.80 o5| | 52893 1.53 1.50 95 „52613 9.1 2.16 45 5261 2.94 2.18 5 „52589 2.31 2.25 35 52486 2.76 2.70 20 „52455 2.90 2.84 20 „52401 3.14 3.08 20 5234 3.42 3.36 5 „52263 3.73 3.67 15 1.52178 4.06 4.00 | 10] | 1.52184 4.04 3.98 10 37* RUBIDIUM. CAESIUM. n (Langley) u (Rubens), 4 u (Langley) # (Rubens), J 4.53733 0.744 0.742 42 | | 4.53566 0.803 0.801 KAD „536 0.782 0.779 450 „5451 0.855 0.851 250 .5359 0.795 0.792 300 „53975 0.895 0.891 200 „9332 0.93 0.92 10 Md 0.920 0.914 70 .53202 1.01 1.00 35 „53202 1.01 1.00 90 „5309 aad 440 10 „52902 1.37 1.35 70 ‚52912 ASD 1.33 200 „52846 1.48 1.45 80 | .52830 1.49 | RED ler CDDB, OS Te 1.70 5 „72597 2.28 2.22 20 | „5264 2.08 2.03 5 52471 | 2.80 2.73 | 25 | 5257 | 2.44 2.35 5 1.52186 | 4.03 | 3.97 | 40 52433 | 3.00 2.93 | 50 iG 2315 Oni 3.45 30 | 1.52208 | 397 3.91 | 40 post MERCURY. n »(Langley)|“ (Rubens)) /* , 4.53198 | 4.01 1.00 | 28 | „53076 dels 4-44 8 „52907 4-30 1.34 | 52828 | 1.52 1.49 Di | | | 1.59759 | 4:70 1.66 5 | * The intensity of the green and yellow mercury lines has been put — 10. Mathematics. — “On the locus of the pairs of common points and the envelope of the common chords of the curves of three pencils.” 2°¢ part.: Application to pencils of conics. By Dr. F. Serum. (Communicated by Prof. P. H. Scroure.) (Communicated in the meeting of December 29, 1906). 9. If the pencils of curves are pencils of conics (r= s = t= 2) then in the case of there being no common base-points the locus is of order fifteen and the envelope of class six. In the following we ( 549 ) wish to treat the case more closely, that one of the pencils has two points in common with each of the two others, where we shall attain at results in another way, which will prove to agree to the general ones and complete these in some parts. Let ABCD, ABEF and CDGH be the three pencils of conies. On one conic of the pencil ABCD the two other pencils describe two quadratic involutions of which the connecting lines of the pairs of points pass through a point A of ZF, resp. a point L of GH. The pair of common points PP” of these two involutions is thus deter- mined by the right line KL. If the conic ABCD describes the whole pencil, A and ZL describe projective series of points on Zand GH. For, if we take A arbitrarily on HF’, the conic ABCD is determined by it, as it must pass through the second point of intersection of CK with the conic ABHFC (as likewise through the second point of intersection of DK and the conic ABEFD); by the conic ABCD the point Z is unequivocally determined. Reversely to a point £ of GH now corresponds one point A. The projective series of points are however in general not perspective; so the line KL or PP’ envelops a conic N touching EF and GH. Of that conic three other tangents are easy to construct, namely by taking for the conic ABCD in succession each of the three degenerations. If that conic is AB.CD then the movable points of intersection with conics of the pencil ABEF lie on CD so that A lies on CD, thus in the point of intersection A, of CD and EF; likewise does Z coincide with the point of intersection L, of AB and GH. The line A,Z, is thus tangent to WV. The construction becomes a little less simple if we take one of the other degenerations eg. AC. BD. By cutting this by the degenerated conic AL. BF of pencil ABLE it is evident that A coincides with the point of intersection of ZF with the line connecting the point of intersection of AF and BD with the point of intersection of BF and AC; in similar manner £ is found. To the locus of the points P and P’ belongs the locus of the points of intersection of the conies of the pencil ABCD with the projectively related series of tangents AL of the conic N. This locus (as is easily evident out of the points of intersection with an arbitrary right line or with an arbitrary conic of the pencil ABCD) is of order five with double points in A, B, C and D; further it passes through Z,F,G and H, as K coincides with £ when the conic ABCD passes through M, ete. If we take for the conic of the pencil ABCD the degeneration AB.CD, then KL passes into K,L, which line cuts the conic AB. CD in the points A, and L,, which thus ( 550 ) lie on the locus of the points of intersection too. By taking the two other degenerations we find four more points of C,. Altogether there are 10 single and 4 double points by which C, is determined. If we take the degeneration AB.CD, the particularity occurs, that the pair of points of the involution described by the pencil ABEF can become indefinite on AB, if namely the conic ABEF breaks up into AB. EF. By this the whole line AB (and of course the line CD too) will belong to the locus proper of P and P’’). To the part proper of the envelope of the lines PP’ the pairs of points PP’ lying on AB or CD contribute nothing but the lines AB and CD (which belong also to the part improper of the envelope, the points A, B, C and D), which does not give rise to a higher class. So the locus proper of P and P”’ consists of the lines AB and CD and the curve C, and is thus in accordance to the general results of order seven. The line A/(CD) intersects C, in the points A and B (C and JD) to be counted double and in L, (Kj). The curve C, has three double points differing from the base-points (of which E, F, G and H are single and A, B, C and D threefold points of C,) namely K,, L, and the point of intersection T of AB and CD. These form a triplet of double points belonging together of which we spoke in $5. The conics of the three pencils passing through one of those double points, also pass through the two others; these conics are AB. CD, AB. EF and CD.GH. To the branches TAK, and TL, of C, passing through 7’ correspond respectively the branches A,7’ and Z,7' passing through A, and Z,, whilst the branches of C, passing through A, and “4, correspond mutually. Summing up we find: For the conics ABCD, ABEF and CDGH the locus proper of the pairs of common points PP’ consists of the lines AB and CD and a curve of order five, having in A, B, C and D double points and in FE, F, G and H single points and further passing through the point of intersection K, of CD and EF and the point of iter- section L, of AB and GH. The envelope proper of the lines PP’ is a conic touching the lines EF, GH and K,L,. 10: If the pomts A,B,C, D,E and Flies ond comic, me latter then belongs to the locus, so that the C, breaks up into that conic and a C, passing through A, B, C, D, G, H, K, and L,. To each conic of the pencil ABCD now belongs the same point A, namely K,, as is immediately evident when we make the conic of 1) More generally: if two base-points of one pencil lie with two base-points of another pencil on a right line, that line belongs to the locus proper. (551 ) the pencil ABHF to pass through C and D. If we take ABCDEF for the conic of the pencil ABCD, then K is indefinite on EF, whilst point Z is to be found somewhere in L, on GH. The corre- spondence between the points A and Z is of such a kind that to a point Z differing from ZL, the same point A always corresponds, namely A, whilst when Z coincides with L, point A’ is arbitrary on HF. So the conic N breaks up into the two points K, and L,. The relation between the conics of the pencil ABCD and the tangents KE or PP’ of N is of such a kind, that to the conic ABCDEF every line through ZL, corresponds and that, for the rest, between the conics ABCD and the lines through A, a projective relation exists, in which to the conics ABCDEF, ABCDG, ABCDH and the degenerated conic AB.CD respectively A,L,, A,G, KH and K,L, correspond. From this is also evident, that the curve C, breaks up into the conic ABCDEF and a C, passing through A, B, C, D, G, H, K, and L, and farther that C, passes through the points of intersection of K,L, with the conc ABCDEF. The double points of C, = AB.CD.ABCDEF-. C, differing from the base-points are A,, L,, 7’ and the two points of intersection of KL, with ABCDEF. The latter two doublepoints do not furnish a triplet of points through which conics of the three pencils pass, but two coinciding pairs of points; the branches through one doublepoint correspond to the branches through the other and, it goes without saying, in such a way that the branches belonging to (, corre- spond mutually and likewise the branches belonging to the conic ABCDEF. 11. If moreover the points A, B,C, D,G and H lhe on a conic, C, breaks up into that conic and the line A,L, (4, then coincides with Z,) so that the locus proper then consists of the conics ABCDEF and ABCDGH and the lines AB, CD and K,L,. When conic ABCD does not pass through /, and / neither through Gand , the point K coincides with A, and / with Z,; so that the pair of points PP’ lying on that conic is always determined by the same line K,/,. Hence A,L, forms part of the locus. The C, has now seven double points differing from the base-points, namely one triplet A,, L,, 7, and two pairs, the two points of intersection of AL, with the conic ABCDEF and those with the conic ABCDGH. If the point A, coincides with L, and therefore also with 7’, i.o.w. if the four lines AB, CD, EF and GH pass through one point, on each conic of the pencil ABCD the two involutions coincide. The locus proper then becomes indefinite. If we bring through an arbi- ( 552 ) trary point P a conic of each of the pencils, then those comics have another second common point, namely the second point of intersection of the line TP with the conic ABCDP. The envelope proper is then still definite and consists of two coinciding points 7. 12. Af the points EF and G coincide, then if the conic of the pencil ABCD passes through / the point K as well as the point £ coincides with /. The series of points K and Z are perspective, the lines AL all pass through a selfsame point U. The conic NV breaks up into two points / and U. As LF belongs to the part improper of the envelope, the envelope proper now consists only of point U. By taking for the conic of the pencil ABCD the degeneration AB.CD it is evident that U lies on the line K,Z,. Another line AZ and by that the point U itself can be constructed in the way indicated in § 9 by allowing the conic ABCD to break up into AC. BD or AD. BC. Between the lines AZ or PP’ through U and the conics of the pencil ABCD exists a projective correspondence, where to the conies ABCDE, ABCDF, ABCDH and AB. CD respectively the lines UE, UF, UH and K,L, correspond. The locus of the points of intersection is a cubic through the points A, B,C, D, U, FE, F, H, K, and L,, which is determined by these 10 points; the third points of intersection of that curve with AC, AD, BC and LD are easy to construct. On the conic ALCDE the two involutions coincide, so that that conic has separated from the (, of $ 9 and has become improper. The locus proper consists now of the lines AB and CD and the above-named C,, so it is of order five. Differing from the base- points the C, has three double points, A,, 4, and 7’ (the point of intersection of AS and CD) forming a triplet. If moreover the points A, B,C, D,E and F le on a conic, no other particularity appears than the point U coinciding with K,. Of the three points of intersection A,, 2, and U of A, L, with the C, the points A, and U now coincide, so that the C, touches the line K, L, in K,. In comparison with § 10 the particularity that appears is this that the point L, coincides with / whilst the pencil of rays L, has passed into the part improper of the envelope and the conic ABCDEF into the part improper of the locus. 13. The case treated in the preceding paragraph is of course not the only one in which the series of points A and / are perspective, the condition of that perspectivity being single, the condition of the ( 553 ) coincidence of / and G being double. The condition of perspectivity can be found out of the condition that the point of intersection |” of HE and GH (as point A’) corresponds to itself (as point /). Now the conic ABCD belonging to V (as point A’) passes through the second point of intersection W of CV with the conic ABEFU, whilst C,W is a pair of points of the involution described on the conic ABCDIV by the pencil ABLF. If this pair of points also belongs to the involution described on that same conic by the pencil CDGH, the point L coincides evidently with V. So this is the case when the cone of the pencil CDGH touching the conic ABCDW in C passes through W. This condition for the perspec- tivity of the series of points A and £ (where of course it must be possible to interchange C with D and likewise AB resp. LF with CD resp. GH) is evidently satisfied when / and G coincide. If U is the centre of perspectivity, there exists between the rays of the pencil U and the conics of the pencil ABCD a projective correspondence, where to the conics ABCD, ALCDF, ABCDG, ALCDH, ALCDW and AB.CD correspond respectively the rays UE, UF, UG, UH, UV and K, L,, whilst moreover to the conic ABCDW all the rays of the pencil V correspond. So the C, of $ 9 breaks np into the conic ABCD, still belonging to the part proper of the locus, and a C, passing through the points A, b, C, D,U, L, HGH, wand J: cutting, the conic in. .A, B,C and D and, the two points of intersection of UV with that conic. The locus proper is thus a C, consisting of the lines AB and CD, the conic ALCDIW and the C, before mentioned. This C, has five double points differing from the base-points, namely, the triplet A,, L,, 7 and the pair formed by the points of intersection of UV with the conic ABCD W. The C, is determined by the ten points, A; B, C, D, E, F, G, H, K, and £, so these ten points will have to lie on a C, if the above condition for the perspectivity is satisfied, and reversely it is easy to prove that when those ten points lie on a C, the series of points are perspective. Suppose namely that the series of points were not perspective. Then it would be possible by keeping the points A, B,C, D, KE, F and G to construct on the line GH (thus by keeping the points /,, 4,, V and J’) by means of the former condition for perspectivity a point H’ in such a manner that the series of points A and ZL. are perspective; H’ is then the second point of intersection of VG with the conic through C, D, Gand W, touching the conic ALCDW in C. So now the ten points A, B, C, D, EF, G, K,, L, and H’ will lie on a C,, however already determined by the Y 7 ( 554 ) nine former points') and thus the same as C, through the ten points A, B,C, D, E, F, G, K,, L, and H. The line VG would then however have four points G, L,, H and H’ in common with this (C,. So we arrive at the following simple result: If the ten points A, B,C, D, EH, F,G,H, K, and L, lie on the same cubic, the series of points K and L are perspective, whilst the centre of perspectivity coincides with the third point of intersection U of K,L, with Cy. The envelope proper breaks up into the point U and the point of intersection V of EF and GH. The locus proper consists of the lines AB and CD, the cubic just mentioned and the conic through A, B,C, D and the two points, in which the right line UV intersects moreover the C, besides in U. If W and G coincide, we immediately see that the above condition is satisfied. The point V lies then in point 7 so that one point of intersection of UV with C, differing from U becomes the point £; the indicated conic is thus the conic ABCDE, which now however belongs to the part improper of the locus. 14. If G coincides with EH and H with F, then the series of points A and Z are connective with double points in MW and F. The pair of points PP’ on an arbitrary conic of the pencil ABCD is now continually described by the same line ZF, thus belonging to the locus proper. If the conic passes through / or /” the two involutions coincide, so that the conics ABCDE and ABCDF belong to the locus, but to the part improper of it. Moreover the lines A and CD belong to the locus proper, so that the latter consists of the three lines AB, CD and EF. An envelope proper is no more at hand, the line connecting P and P’ coinciding with Ab, CD or EF when P and P’ differ from the base-points. In comparison with $ 12 the particularity appears that U coincides with /’, that the pencil of rays U passes into the part improper of the envelope and that the C, breaks up into the conie ABCDF becoming improper and the right line ZF. The case of the pencils of conics ABCD, ABEF and CDEF can be profitably used to define with the help of the principle of the permanency of the number the order of the locus of P and P’ and the class of the envelope of PP’ for the case of pencils of conics lying arbitrarily with respect to each other. Starting from this simplest 1) The C} is only then not determined by these nine points if two of those points coincide in such a way that the connecting line is indefinite (e.g. G with E or K, with Lj). Then the ten points lie on a C3, whilst it is easy to prove that the correspondence between K and L is perspective. (555 ) case, it is easy to reason that PP’ coincides with AB, CD or ZF and so the locus proper consists of these three lines and there is no envelope proper. The part improper of the locus however consists of six conics ABCDE, ABCDF, ABEFC, ABEFD, CDEFA and CDEFB, the part improper of the envelope of the six points A, B,C, D,E and F. The total locus is thus of order fifteen, the total envelope of class six, so that for arbitrary position of the pencils of conics this same holds for the locus proper and the envelope proper. Sneek, Nov. 1906. Mathematics. — “The locus of the pairs of common points of four pencils of surfaces.’ By Dr. F. Scuun. (Communicated by Prof. Po Ht. ScHovurs): (Communicated in the meeting of December 29, 1906). 1. Given four pencils of surfaces (PF), (ff), () and (F,) respect- ively of order 7,s,¢ and uw. The base-curves of those pencils can have common points or they can in part coincide, in consequence of which of three arbitrary surfaces of the pencils (4), (/) and (/,) the number of points of intersection differing from the base-curves can become less than stuw; we call this number a, calling it 5 for the pencils (/;),(/,) and (/’,), c for the pencils (#,) (/;) and (F,) and d for the pencils (/,), (/,) and (£). We now put the question: What ws the order of the surface formed by the pairs of points P and P’, through which a surface of each of the four pencils is possible ? If the points P? and /P” do not lie on the base-curves we call the locus formed by those points the locus proper L on which of course still eurves of points ? may lie for which the corresponding point P’ lies on one of the base-curves. If one triplet of pencils furnishes at least several points of intersection which are situated for all sur- faces of those pencils on one of the base-curves, then there is a surface that does satisfy the question but in such a manner that if we assume /? arbitrarily on this surface the point 7” belonging to it is to be found on one of the base-curves; this surface we call the part improper of the locus, whilst both surfaces together are called the total locus. 2. To determine the order # of the locus proper L we find the points of intersection with an arbitrary right line /. On / we take ( 556 ) an arbitrary point Qy, and we bring through that point surfaces F.. F, and F, of the pencils (7), (4%) and (F,). Through each of the «— 1 points of intersection of those surfaces not situated on the base-curves of those surfaces we bring a surface £,. These a —1 surfaces /. intersect the right line / together in (a—1)r points Q,, which we make to correspond to the point Qj. The coincidences of this correspondence are: 15t the points Qs, determining four surfaces which intersect one another once more in a point not lying on the base-curves, thus the 7 points of intersection with the surface L, 2rd the points of intersection with the surface fi, belonging to the pencils (/), (#) and (45), the loeus of the points S determining three surfaces whose tangential planes in S pass through one line. To find the number of coincidences we have to determine the number of points Q., corresponding to an arbitrary point Q, of £. To this end we take on / a point Q,, arbitrarily and bring through it an HF, and an 4. Through each of the 4 points of intersection of these surfaces with the surface # through ( (not lying on the base-curves) we bring an #, which 5 surfaces I’, intersect together the line 7 in ds points Q, which we make to correspond to Qt To find the number of points Q,, corresponding to an arbitrary point Q, of / we take Q, arbitrarily on /, we bring through Q, an Fand through @, an /, and through each of the c points of inter- section of those surfaces with /, an #,, which furnish c surfaces F,, cutting / in ct points Q,; reversely to Q, belong du points Q,, so that we find between the points Q, and Q, a (ct, du)-correspond- ence, of which the cf+du coincidences give the points Q,, belong- ing to the point Qs. So between the points Qin and (}, exists a (bs, ct + du)-correspondence, of which the coincidences consist of the + points of intersection of / with the surface /, through Q, and of the points Qs, corresponding to Q,.; the number of these thus amounts to bs + ct 4 du—r. So between the points Q,,, and Q. there is an (ar—r, bs+-ct+du—)- correspondence with ar bs ct-+-du— 2r coincidences. To find out of this the number of points Qs: we must first determine the order of the surface Pts. This surface may be regarded as the surface of contact of the surfaces of the pencil (2) with the movable curves of intersections (1, of the surfaces of the pencils (4) and (Fo) '). So the question is: 1) We shal! call this surface the surface of contact of the three pencils meaning by this that in a point of this “surface of contact” the surfaces of the pencils, though not touching one another, admit of a common tangent. ( 557 ) 3. To determine the order of the surface of contact of a twofold infimte system of twisted curves and a singly infinite system of surfaces. To this end we shall first suppose the two systems to be arbitrary. To determine the order of the surface of contact we count its points of intersection with an arbitrary right line /. To this end we consider the envelope H, of the oe? tangential planes of the curves of the system in their points of intersection with / and the envelope E, of the oo! tangential planes of the surfaces of the system in their points of intersection with 7. The common tangential planes not passing through / of both envelopes indicate by means of their points of intersection with / the points of intersection of / with the surface of contact. In order to find the class of the envelope /, (formed by the tangential planes of a regulus with / as directrix) we determine the class of the cone enveloped by the tangential planes passing through an arbitrary point Q of /. If the system of curves is such that g curves pass through an arbitrary point and w curves touch a given plane in a point of a given right line, the tangential planes of E, through Q envelope the p tangents in Q of the curves of the system through Q, and the line / counting W times; for each plane through / is to be regarded yp times as tangential plane, there being y curves of the system cutting / and having a tangent situated in this plane. The envelope LL, is thus of class p + yw and has 1 as y-fold line). To find the class of the envelope #, we determine the number of its tangential planes through an arbitrary point Q of /. If now the system has gu surfaces through a given point and rv surfaces touching a given right line, the tangential planes of the envelope passing through Q are the tangential planes in Q to the u surfaces passing through Q and the tangential planes of the » surfaces touching /. So the envelope B, is of class ud-v with v tangential planes through 1. Hence both envelopes have (p +) (u +r) common tangential planes. Each of the » tangential planes of #, passing through / is however a y-fold tangential plane of #, and so it counts for w common tangential planes. So for the number of common tangential planes not passing through /, thus the number of points of intersection of / with the surface of contact we find: (y+ +>) — Dr = gr + m+ gu, therefore : 1) The regulus as locus of points has however line 7 as ¢-fold line. ( 558 ) The surface of contact of a system (p‚ W) of cw twisted curves *) and a system (u, v) of op! surfaces”) is of order gv + wu + pu *). 4. To determine the order of the surface of contact*) of the systems #,,”,) , (u,v) and (u,,»,) each of o' surfaces, we regard the system (y,w) of the curves of intersection of the systems (u, , v,) and (u, ,r,). Of these curves of intersection u‚u, pass through a given point, so p=uu,. The w points, where the curves of intersection touch a given plane in a point of a given right line, are the points of inter- section of that given line with the curve of contact of the systems (u,,v,)*°) and (u,,»,) of plane curves, according to which the given plane intersects the systems of surfaces (u,,»v,) and (u, ,r,\. This curve of contact is of order u,v, + ur, + wu, thus: : y= ur, + HP, sik u‚l,- The surface of contact to be found is thus the surface of contact of a system (uu, tv, + HoV, + Us) Of oo° twisted curves and a system (u,,»”,) Of oo! surfaces, so that we find: The surface of contact of three systems (u, ‚v‚), (u, , v‚) and (u, , »,) of oo! surfaces is of order UP, HUP, HMG PF 2M iat - If the three systems are the pencils (#5), (7) and (#) we have = =u, =1, p= sl) Pp, SES), Value So we find: The surface of contact Far, of the three pencils of surfaces (Fs), : (F,) and (F) is of order ') System with ? curves through a given point and y curves cutting a given line and touching in the point of intersection a given plane through that line. 2) System with » surfaces through a given point and » surfaces touching a given right line. 3) This result is also immediately deducible from the Scuuzerr formula cp?= p8.G+p'g'e. p'ge+ p® .pge (Kalkül der abzählenden Geometrie, formula 13, page 292) for the number of common elements with a point lying on a given line of a system ©’ of 3 and a system © of oet right lines with a point on it. If we take for ©’ the tangents with point of contact of the system of curves (7, ¥) and for = the tangents with point of contact of the system of surfaces (y ,v), then Poke 3 PGs Hh ie, ee whilst zp? is the order of the surface of contact. 4) Locus of the points, where the surfaces of the three systems have a common tangent. 5) System of cel curves of which wg, pass through a given point and », touch a given right line. said eee ( 559 ) 2(s +-¢+ u — 2). 5. To return to the question which gave rise to the preceding considerations we find for the number of points Q,.;, on the arbitrary line /,- which are the points of intersection of / with the locus proper L: ar + bs + ct + du — 2r — 2(s+t+u—2)= =art+ bs +e+du—2(r4+s+t¢+u)44. So we find: The locus L of the pairs consisting of two movable points common "io a surface out of each of the pencils (F,), (Fs), (Fi) and (F,) of orders 7, s, t and u, and not lying on the base-curves, isa surface of order ar + bs + et + du—2(r+s+t+4+ u) + 4. Here a is the number of points of intersection not necessarily situated on the base-curves of the pencils (F), (F.) and (F,); b the analogous number for the pencils F5), A) and (Fo), etc. 6. It the pencils have an arbitrary situation with respect to each other, then a=stu, etc, so that then the order of the locus becomes 4(rstu + 1) — 2(r Hs tt wv). ‘That order is lowered when three of the base-curves have a common point or two of the base-curves have a common part, which lowering of the order can be explained by separation as long as the total locus is definite, i.e. as long as the four base-curves have no common point and no triplet of base-curves have a common part. For, if As is a common point of four base-curves then the surfaces of the four pencils passing through an entirely arbitrary point P have another second point in common, namely As; if By, is a curve forming part of the base-curves B, B, and B, of the pencils U), (4) and 1), then the surfaces of the pencils passing through an arbitrary point P have moreover the points of intersection in common of By, with the surface F, through P; so in both cases the arbitrary point P belongs to the total locus. If the basecurves B, B, and B, have a common point A, then on account of that point the number « is diminished by unity without having any influence on 6, e and d. The order of Z is thus lowered by 7 on account of it, which is immediately explained by the fact that the surface EF, passing through Agy separates itself from the locus. ( 560 ) If the base-curves B, and B, have a curve B,, in common of which for convenience we suppose that it does not intersect the base-curves B, and B, this B,, has no influence on c and d, whilst a is lowered with sm and 6 with rim, where im represents the order of the curve B; for, when F,, F, and #, are three arbitrary surfaces always sm points of intersection lie on B. The order of Lis thus lowered with 2rsm by B. This can be explained by, the fact, that the locus of the curves of intersection Cs of surfaces F, and F, passing through a selfsame point of By") separates itself from the locus of P and P'. That the locus of those curves of inter- section is really of order 27sm is easily evident from the points of intersection with an arbitrary line / We can bring through an arbitrary point Q, of lan F cutting Ay, in rm points; through each of those points of intersection we bring an /,, which rm sur- faces F, cut the right line / in rsm points Qs. To @, correspond rsm points Q, and reversely. The 2rsm coincidences are the points of intersection of / with the locus of the curves of intersection C\.. 7. The base-curves B, B, B, and B, of the pencils are morefold curves of the surface ZL. If A, is a point of B, but not of the other base-curves, then A, is an (a — 1)-fold point of £. For, the surfaces As, and /, through A, intersect one another in a—1 points, not lying on the base-curves, each of which points furnishes together with A, a pair of points satisfying the question. Each point of B, is thus an (a —1)-fold point, i.o. w. B, is (a—1)-fold curve of the surface L. Let A,; be a point of intersection of the base-curves B, and By, but not a point of B, and B. An arbitrary point P of the curve of intersection C), of the surfaces /, and /, through A, furnishes now together with A,; a pair of points PP’ satisfying the question pro- perly, as A, is for each triplet of pencils a movable point of inter- section not lying on the base-curves. If we let P describe the curve Ci, then the tangent /-, in A,, to the curve of intersection of the surfaces /, and F, through P describes the cone of contact of Z in the conic point A,,. The tangents m, and m, in A,sto B, and Bs are (a—1)- resp. (6—1)-fold edges of the cone. This cone is cut by the plane through m, and ms only according to the line m, counting (a—1)-times and the line m, counting (6— 1)-times, as another line /,, lying in this plane would determine two surfaces 1) If Br cuts the curve Bs in a point Astu, then the surface HF passing through Astu separates itself from the locus of the curves of intersection Crs. r ( 561 ) F, and F, touching each other in A,,, whose curve of intersection, however, does not cut the curve Cy. The tangential cone of L in A, is thus of order a Hb — 2"). Let Aw be a point of a common part Bs of the base-curves B, and B, but not a point of 5, and 5, We get a pair of points PP’ Sages, 1) with a point ?’ coinciding with A, when the surfaces fF, and F, : 1) 8 have in A‘! a common tangential plane VV, and pass through a selfsame point P of the curve of intersection C;,,of the surfaces Pand Ff, through A. If we let P describe the curve C}, , then on account of that between the planes VV, and JV, touching in A‘? the surfaces F, and F, through P, a correspondence is arranged, where to V, correspond /—1 planes V, and to Vs correspond a —1 planes V,. One of the a+ — 2 planes of coincidences is the plane through a : the tangents in A,; to B, and C,; this plane furnishes no plane V,;. The remaining a + 6 — 8 planes of coincidence are planes V, and indicate the tangential planes in A to the surface L. So Br is an (a +b—3)-fold curve of L. 8. Let us then consider a common point A of the base-curves B,, B; and B,. We get a pair of points PP’ with a point P’ coin- ciding with A, when the tangential planes in A, to F, Fsand F, pass through one line /; and these surfaces intersect one another again in a point P of the surface /, passing through A There are o' such lines /,,,, forming the tangential cone of Z in point As. The tangents m,,m, and m in A, to B,, B, and B; are (a — 1)-, (6—1)- and (c —1)-fold edges of that cone. So the plane through m, and mm, furnishes a@—-+6—2 lines of intersection with the cone coinciding with m, and m,. Moreover c —2 other lines Lt lie in this plane. For, the surfaces F, and /’, touching this plane intersect /, in c—2 points not lying on the base-curves; the surfaces /, through those points intersect the plane through 1, and mm, according to curves whose tangents in A, are the mentioned 1) The order of this cone can also be found out of the: number of lines of intersection with an arbitrary plane « through 4,;. If l, and J, are the lines of intersection of « with the tangential planes in A,; to the surfaces F, and F through P, then to J, correspond b—1 lines 7; and to /; correspond a—1 lines /,, so that in the plane = lie a+b — 2 lines ls. 38 Proceedings Royal Acad. Amsterdam. Vol. 1X. ( 562 ) i} lines /,;. So the tangential cone of Lin A,st is of order a+ b+ c— 4). B re B 1) 5 A point of intersection Als of B, with a common part A, of the base-curves B, and B, is a conic point of L, the tangential cone of which is formed as in the previous case by co’ lines /,.;. The tangents m, and m4 in AS to B, and By, are (a—1) and (b + c — 3)-fold edges of that cone. As no other lines /,,; lie in the plane through : tel : ; , DR m, and ms, it is evident that the tangential cone of Lin AY is like- wise of order at+b+c—4 ’). Let AS be a point of a common part B of the base-curves B,, B, and B,. The point P’ of the pair of points PP’ coincides with (9 . 7 7 . (2 As: When the surfaces F,, Hs and F, have in A; the same tangen- tial plane WV, and cut one another in another point Pof the surface 4 2 Wert. HF, through AS, If we now consider an F; and an 4 having in (2) u A the same tangential plane V‚, and if we consider through each of the c—1 points of intersection of #, F, and F, not lying on the base-curves an / of which we indicate the tangential plane in AS by V, then to V‚s correspond c—-1 planes V; and to V; cor- respond a + 6—1 planes WV, (as for given V; a (4, a)-correspondence exists between WV, and V, of which WV, is one of the planes of coin- cidence). Among the atb + c¢— 2 planes of coincidence Vs, V; there are however three which give no plane Vs, namely the planes V.;, for which the corresponding surfaces / and #, furnish with F, three points of intersection coinciding with Aj. For this is neces- - (9 7 : sary that /’, touches in Aj the movable intersection of /, and F. Now the tangents of those intersections for all surfaces F, ' (3) > and F, touching each other in A;,; form a cubic cone having for B 5 2) double edge the tangent ms to Bt in point Ae *). This cone is cut by the tangential plane in AS) to F’, according to three lines, furnishing with 7, planes V, which are planes of coincidence 1) This order can also be determined out of the number of lines /,s¢ in a plane e passing through A,s. In this plane we finda (c — 1, d + b — 2)-correspondence between lines 7,5 and lines 7; of which however the line of intersection of <« with the tangential plane in A, to Fy is a line of coincidence, but no line ys. 2) This is immediately evident if we take for (F,) a pencil of planes and for (F's) a pencil of quadratic surfaces all passing through the axis B, of the pencil of planes. The cone under consideration then becomes the cone of the generatrices of the quadratic surfaces passing through a given point of B. We can easily convince ourselves that the same result holds for arbitrary pencils of surfaces. ( 563 ) of V,s and V,, but not planes Vs. So there are a-+6-+-c—5 planes 9 Viso, which are the tangential planes of Z in the point A; i. o. w. Brat ts (a +b + ¢ — 5)-fold curve of surface L. 9. We then consider a common point As, of the four base-curves. We get a pair of points PP’ with point /” coinciding with A, when B Fe & and Ff, have in A 2 common tangent Le, and all pass once again through a selfsame point P. The oo! lines /,,;, form the tangential cone of Z in A4. To determine the number of lines [rs in an arbitrary plane e through As, we take in this plane an arbitrary line Jd, through As and we bring through the d—41 points of intersection (not lying on the base-curves) of the surfaces FE, Fs and HF, touching J, the surfaces F,, whose tangential planes in A,s, cut the plane ¢ according to lines, which we shall call /,. To Ls, now correspond ¢—1 lines /, and to /, correspond a+b-+c—2 lines /..,, as there exists between /,., and /, when /, is given a (c‚a + b)-correspondenee, of which /, and the line of intersection of ¢ with the plane through the tangents in A, to B, and B; are lines of coincidence, but not lines /. So there are atbtctd—=s3 lines of coincidence /4 /, of which however three are not lines Zos, The common tangents in As, of the surfaces F,, F, and HF, possessing three points of intersection coinciding with Ast and where therefore the intersection of two of those surfaces shows a contact of order two to the third, form namely a cubic cone ') of which the lines of intersection with ¢ are lines of coincidence but not lines Lost. So in e lie a+6b+c+d—6 lines (psy, i. 0. w. the tangential cone of L in Aysty is of order a 4 b He td—6*). 1) This is again evident when taking for (#,) and (fF) pencils of planes with coplanar axes B, and B, and for (#) a pencil of quadratic surfaces passing through a line containing the point of intersection S of B, and B, The line of intersection of the planes F, and F; shows only then a contact of order two to Fy when that line of intersection lies entirely on F/, so that the cone under consideration becomes again the cone of the generatrices of the quadratic surfaces passing through S. 2) That order can also be found out of the lines of intersection with the plane V;s through the tangents m, and m in A,su to By and B;. Those lines of inter- section are: the line m,, counting (a — 1)-times, the line mm, counting (b — 1)- times and c 4d — 4 other lines. This last amount we find by drawing in plane V,; an arbitrary line 7; through Arsu. The surface F4 touching J, cuts the surfaces F, and F, touching V;s; in d — 1 points (not lying on the base-curves) through which points we bring surfaces F, whose tangential planes in A,s cut the plane V‚s according to lines to be called /,. Between the lines /¢ and 7, we now have a (d — 1, c — 1)-correspondence of which the nodal tangents in Ars of the inter- section of the surfaces fF, and F; touching JV;; are lines of coincidence. The remaining Cc + d — 4 lines of coincidence are lines /;stu. 38* (- 564 ) The preceding considerations hold invariably for a point Al, lying on the base-curves B, and B, and the common part B,, of the base-curves B, and B). In a point of intersection AS of B, and B, the tangential cone is likewise of order a + b 4 c + d — 6 as that cone has the tan- gents mm, and m, to B, and By as (a + 6— 3) and (ce + d —3) fold edges, whilst in the plane through m,; and m,, no otber right lines List are lying. A point of intersection AG), of B, and Bary is also a(a-b+c+d—6) fold point of L as m, and ma, are (a —1)- and (b + ¢ + d—5)- fold edges of the tangential cone and the only lines of intersection of that cone with the plane through m, and msqy. (4). oe ee If finally AM). is a point of a common part Bs of the four base- curves, then the point ?’ of the pair of points PP’ coincides with (4 ’ 7 a] 5 4 AD, when the surfaces F,, F,, /, and #, have in Ars the same tangential plane Vs, and all pass through a same point P. Let us now assume an arbitrary plane Vo, passing through the tangent SN 4) . 1 5 > ; bete in A®, to Bs. The surfaces F,, sand /; touching this plane in AD, cut one another in d—1 points P, through which we bring surfaces J, of which we call the tangential planes in A}, VAA Thus we obtain a correspondence, where to Vs, correspond d—1 planes VV, and reversely to VV, correspond a + 6+ c—1 planes 7st; for when V, is given there is between V,, and V; a (c, a+ 6) correspondence, of which JV, is plane of coincidence, but not a plane Vs. So there are a + b + ce + d—2 planes of coin- cidence VV, of which however jive are not planes Vos. These are namely the tangential planes of the surfaces #, /’; and #, of : : os B bert . (4) B which one more point of intersection coincides with As which 1) [t is also easy to see from the lines of intersection with the plane View through the tangents ms and mma to B, and By that the tangential cone in AD, is of order a db He} d—6. The line ms counts for b — 1 lines of inter- section, the line mm for ¢-+d— 3. Further, the surfaces Fs, HF, and F, touching Ven cut one another in a — 2 points not lying on the base-curves; through those 1 points we bring surfaces F, whose tangential planes in Arsa cut the plane Vsuu along to lines which lie on the tangential cone. ( 565 ) occurs five times *). So there remain « + 6 + ce 4 d — 7 planes V, rstu which are the tangential planes of JZ in the point yobs so that Brsta is a (a +b6+e+d—7) fold curve of L. 10. So we find: Of the locus proper L of the pairs of points P and P’ the base-curve. B, of the pencil (F,) is (a —1)-fold curve, the common part B of the base-curves B, and B; is (a + b — 3)-fold curve, the common part B,s of the base-curves B,, Bs and By is (a + b + ¢—5) fold curve and the common part Bystu of the four base-curves is (a+b+c+d—%)-fold curve. The points of intersection of the base-curves are conic points of L, namely a point of intersection of B, and B, is (a+b — 2)-fold point, a point of intersection of B, Bs and B, or of B, and By is (a+b+¢—-4)-fold point and a point of intersection of B, Bs, Br and B, or of B,, B, and Bu or of Bs and By or of B, and Bey is (a +b + e 4d —6)- fold point. *) 11. The base-curves of the pencils are not the only singular curves of the surface L. There are namely oo! triplets of points lying on a surface of each of the pencils. These triplets of points form a double curve of L. If P, P', P" is such a triplet and if P1 and P2 are the sheets through P of the surface, then the sheets P'1 and P"2 correspond to them. Through P’ passes another sheet P’3 and through P" a sheet P"3 which sheets correspond mutually. The pair of points not lying on the base-curves is movable along the sheets P1, P’1, along the sheets P2, P"2 and along the sheets P’3, P"3; on the base-curve a third point then joins the pair. Further there is still a finite number of quadruples of points, 1) The number five is found in the following way. The tangents of the movable . ° ° . (4) . intersections of surfaces F; and Fy touching each other in As form a cubic cone having the tangent M,stu to Brs as double line. Such an intersection shows to the surface PF, a contact of order two when it touches the movable intersection of F, and HF, so if its tangent in Afl lies on the cubic cone belonging to the pencils (F) and (#4). As this last cone has also mers as double edge, both cones have 9—4=D5 lines of intersection differing from mrs which connected with Mrstu furnish the five planes under consideration. 2) If the total locus is not indefinite, 1. o. w. if there is no point common to the four base-curves then B; is a (stu — 1)-fold curve and B,; a (stu + rtw — 2) fold curve of the total locus whilst a point of intersection of B, and Bs is a (stu + riu — 2)-fold point and a point of intersection of B, Bs and By or of B, and By a (stu + riu + rsu — 3)-fold point of it. ( 566 ) through which passes a surface out of each of the pencils. Through the points P,P’, P" and P” of such a quadruple pass three sheets of the surface Z and three branches of the double curve. The 12 branches of the double curve through those four points we can call PAs PS PAP DBA PL: PO RUE Pa ie ee a way that the triplet of points is movable along the branches ELP ASP"), alone P2; P2, F2 along Fo 3 FS and aoe P'4, P'4, P'"4. If the sheet of Z passing through P1 and P2 is called P12, then the corresponding sheets (i. e. sheets along which the pair of points not lying on the double curve is movable) are ee and F129. PTS. and "sa ete; Geophysics. — “Current-measurements at various depths in the North Sea.’ (First communication). By Prof. C. H. Winp, Ltt. A. F. H. Datauisen and Dr. W. E. Rinecer. In the year 1904 accurate measurements of the currents in the North Sea *) were started by the naval lieutenant A. M. van Roosen- DAAL, at the time detached to the “Rijksinstituut voor het Onderzoek der Zee”, having been proposed and guided by the Dutch delegates to the International Council for the Study of the Sea. By him four apparatus were put to the test, viz. 2 specimens of the current-meter of PeTTERSSON ’*), one of that of Nansen *) and one of that of Exman *), all destined to determine the direction and the velocity of the current at every depth. The experiments were partly made on the light-ship “Haaks”, where Dr. J. P. vaN DER STOK, the Marine Superintendent of the Kon. Nederl. Meteorologisch Instituut, also took part in them. Other experiments were made in the harbour of Nieuwediep and further, from the research-steamer “Wodan”, in the open North Sea at a station (H2) of the Dutch seasonal cruises °), situated at Lat. 53°44’ N. and Long. 4°28’ E. 1) Cons. Perm. Intern. p. l’expl. de la mer, Publications de circonstance No. 26: A. M. van Roosenpaat und C. H. Winp, Prüfung von Strommessern und Strom- messungsversuche in der Nordsee. Copenhague, 1905. 2) Publ. de circ. No. 25. Se ede: io eNO ream. JAAT PEA OH ®) Quarterly cruises of the countries taking part in the international study of the sea, along fixed routes, observations being made at definite points or “stations”. ( 567 ) The apparatus of NANSEN appeared to be unfit for the measure- ments on the North Sea; it was not calculated for the strong tidal currents occurring there (e.g, 60—100 em/sec.), and also the putting out of the apparatus in unfavonrable weather was hardly possible without doing harm to the instrument. In more quiet water, however, it seems to be very useful. The apparatuses of PerrERSSON and EkMAN appeared to be better fit for the observations in the North Sea. Some improvements in the construction were proposed, partly also put into practice, by Van Roo- SENDAAL and Winp, by which the instruments have gained in fitness. For a description of the construction of the current-meters used, and the experience made in using them, we may refer to the publications mentioned. The following few words may be sufficient here. It appeared that pretty large oscillations, e.g. 15° to both sides round the longitudinal axis, did not yet render observation impossible. In 32 out of nearly 200 observations by Van ROOsENDAAL as much as the figure 4 was noted for the motion of the sea, in 40 to 50 cases the oscillations amounted to 10 a 20° to either side, and yet the accuracy and certainty of these measurements were ouly excep- tionally insufficient. In the parallel-observations with the apparatus of PrTTersson and Ekman the agreement in indicating the velocity appeared satisfactory: In one series of 23 measurements e. g. the average difference amounted to 4.8 em/sec, whilst the smallest was 3.1, the greatest 6.3. Nor did the indications of direction, as given by the two instru- ments, show great differences. The observations with EkKMAN's appa- ratus bear to some extent a check in themselves, as, by the construction of the instrument, every observation includes a series of consecutive readings at small intervals. In by far the greater part of the readings- observations these separate did not considerably vary. In 128 cases the direction of the current could be estimated from them : To less than 10° in 105 cases, 10—20 15 20—30 2 30—40 0 40—50 2 more than 50° 4. Compared with the probable direction, as derived from the instru- ment of Exman, that which was determined by means of PerrErson’s instrument deviated : ( 568 ) in 65 cases less than 10 37 10—20 15 20—30 5 30—40 1 40—50 8 more than 50°. Van ROOSENDAAL and Winp took from the whole of observations made at station H, the most probable values direction and velocity of current at the various depths and represented them graphically. They constructed for the different series of observations, each lasting 12 or 24 hours, in the first place central vector-diagrams, by drawing from a fixed point the successively determined currents as radii-vectores and connecting the terminal points by means of straight lines or ofa curve, and in the second place progressive vector-diagrams, by drawing the current- vectors, this time interpolated for the successive full hours, one after and attached to the other. In the first kind of diagrams the periodical currents, and in the second the residual currents make themselves most apparent. _ The measurements were continued at the station H2 during all the following seasonal cruises of the “Rijksinstituut”, first by Van RoosenDAAL and afterwards by the naval lieutenant DALHUISEN, who succeeded the former in his detachment. At the more recent measu- rements the current-meter of EKMAN was always made use of. The following table gives the dates of the series of observations and the number of measurements *). E | En Number of! NO. Time. Measure- C5 peek Observer. | ments. : | Dei Ee keeg 56 | _5,20,35 | EKMAN,. | VAN ROOSENDAAL, till UA om MAD dg 2. [from 7 Nov. ’05 7.48 a.m. | 53 | PLN 5 VAN ROOSENDAAL fill 8 re nam.) 3 | and DALHUISEN. | 3. |from 7 Febr.'06 7.20 p.m. | j | | till Be oe ai 18 >» » » » | DALHUISEN. | | 4. |fram 2 May 06 635 am.| ;- | ES Sr Cn TREE a D4 »» » | » » 1) A more detailed description of these observations forms the contents of the last issue of the “Publications de circonstance’’ No. 36. ( 569 ) At these researches wind and weather were on the whole favour- able; the wind was in a few cases noted 7 at most, at which force, however, the observations had to be put a stop to in Feb- ruary 1906 *). On the plate added, the new measurements are again represented graphically in central and progressive vector-diagrams. Also the central diagrams, have been constructed this time with the aid of values interpolated for full hours, the directly measured values however, having still been indicated by dots. It is principally to give a full idea of the variability in direction and velocity of the currents, that these diagrams of the new series of observations have been reproduced fully here. Comparing the values of the velocity near the surface and in the depth, we see that in 3 out of the 4 cases they show a rather distinct decrease at an increase of depth. Also at the former series of observations at H2 (8—4 Aug., 8—9 Aug. and 2—3 Noy. 1905 °), also 8—9 Febr. 1905*)) the same result was arrived at. Also differences of phase in the periodical currents are noticed in most cases between the surface and the depth, though a distinct law may not immediately be obvious here. The striking difference in amplitude of the tidal currents during the observations in August 1905 and February 1906 on the one side and that of November 1905 and May 1906 on the other, is certainly connected with the age of the tide, as it was with the first nearly spring-tide (15'/, and 14 days after N.M.), with the last nearer to dead neap (10 and O days after N.M.). The small number of series of observations that can be disposed of, does of course not allow at all to already think of a calculation of tidal constants, nor to give a correct description of the average variation of the currents. The unmistakable general agreement, however, between the different current-diagrams justifies sufficiently an attempt to compose them. As no doubt moon-tide will have played 1) The reliability of the new observations is no doubt greater than that of the former, if we take into consideration, that in August and November 1905 and in February and May 1906 the Wodan lay moored, so that her motion was conside- rably smaller than on the former occasions, when she had cast only one anchor. It may still be mentioned that an experimental and theoretical investigation was started about the influence of the movements of the ship upon the indications of the current-meter, which, however, has not yet led to a satisfactory result. 2) Publ. de Circ. NO, 26. Brie. (a. eo 4 eee OO: (570 ) the principal part, we have thought best for this purpose to compose for the successive full moon-hours the current-values as they follow by interpolation from the different diagrams. The averages thus obtained have been combined in new diagrams, which are represented on the plate, in the last column of figures, and that by black curved lines. In order to complete the matter and to allow comparisons, in the same way average diagrams have been derived from the observations made in the past year at H2 (see above) and represented in the same figures on the plate by black-and-white curves. The arrows drawn in these figures indicate: in the central dia- grams the direction of the current at the moon’s transit, in the progressive diagrams the total residual current during a half moon-day. A comparison of the average current-diagrams for various depths or also of the newer with the older ones might give rise to all kinds of remarks. With a view to the small number of data, how- ever, on which the diagrams are based, it would perhaps be incon- siderate to mention all of them here. We therefore confine ourselves to what follows. Difference in Phase of the tide at different depths. August 1905 —May 1906 August—November 1904 20 M.—5 M. 35 M.—20 M. | 20 M —70 M. | 30 M.—20 M. C Transit 18° | 5e | 0 | 3° one hour after » 24 | — 6 | —13 14 2 23 — 3 — 8 5 3 20 5 Livia | 6 4 25 HR) | 5 14 5 26 | 0 18 — 5 6 25 | 8 | 3 22 5 » before » 19 17 0 47 4 25 15 — 9 47 3 8 — 6 — 3 25 2 4 10 — 6 24 1 6 che — 6 At | Average 13°85! 4°30! . — 1°50! 42°45 (57) The tidal eurve shows not only at different depths, but also in the older and newer observations, generally the same shape. Its size, on the other hand, both in the older and more recent observations, appears to be smaller near the bottom than near the surface. Also its orientation and the situation of the point in it, which relates to the moment of the moon’s transit, or, more generally, the phase of the tidal current, seems to change in a definite sense as the depth increases. This last relation may be specially illustrated by the following table. It appears from the table, that the tide is on the whole accelerated in the depth, compared with higher layers; but the table also proves that the phenomenon underlies varying influences, besides constant causes, among which perhaps may be reckonned the shape of the bottom of the sea and ihe rotation of the earth. The residual current is by no means constant; at the new obser- vations it has been much stronger than at the old; it shows consid- erable fluctuations also, when the progressive diagrams of the different days of observation are compared. At the new observations this residual current was on an average stronger near the surface than in deeper layers. This particular may perhaps be principally attributed to the action of persisting winds, which at least on the observations of August 1905 and May 1906 had a very marked influence, rendered quite obvious by the special diagrams for these dates. The figures for the residual current as deduced from the newer observations are the following: Depth. Direction. | Velocity. 5 M. N 304° E 1/4 mile p. hour 20 317 Is i 35 309° Io, as deduced from the older: 71) M. N 319° E | Yyg mile p. hour 20 295° 1/26 30 | 323° Iig. These results are worth comparing with the following table of values for the year-average of the residual current at the Noord- Hinder (Lat. 51°35'.5N., Long. 2°37 E.), calculated by VAN DER STOK ”) from current-estimations near the surface during five consecutive years. 1) Average of depths of 1, 4, 5, 6, 10 M.; at a depth of 35 M. measurements were made by Van Roosenpaat only in February 1905. 2) J. P. van per Srox, Etudes des Phénomènes de Marée sur les côtes néer- landaises ; Kon. Ned. Met. Inst. No. 90, Il. p. 67, 1905. (572) Year, | Direction. | Velocity. | | 1890 N 16° E 0.024 miles p. hour. 91 15 62 92 16 35 93 29 47 94 27 47 Average | N 2° E | 0,044 miles p. hour. Here it appears that the average residual current, which — as we mention in passing — has at this point quite another direction than at H2, even from year to year does not at all remain constant in strength which may perhaps be an indication for differences in the quantity of Atlantic water, entering through the English Channel from year to year. The question may be put, whether and how far the results attained by the current-measurements described, deviate from what is known from the charts, in general use, about the currents near the station H2, The subjoined table allows of a comparison with statements, bor- rowed from a chart, published by the British Admiralty *), and shows | From the Charts. Obser ved. Hour. Direction. | eerd Direction. | fant | | 5 before H.W.Dover, N 90° E 03-02 | N 73° E | 0,3 I 110 05—0,3 | 115 0,4 3 135 0,9—08 | 147 | 0,4 2 | 160 | 06—04 | 189 0,3 1 | 180 0,3—0.2 | 227 0.4 H. W. Dover | — O | 266 0.5 1 after H.W. Dover. 260 0302 .| 280 05 2 | 300 | 0604 | 296 0,6 3 | 300 ay ETE e ae 331 05 A 315 0604 342 hee 5 0 0.3—0,2 | 9 | 0,4 6 | 50 iS; irt ee zl 40 Kol ens 5 1) Tidal Streams North Sea 1899. A. F H DALHUISEN, W. E‚ RINGER and C. 1905, August 16—I7. fi , \ e+ \\ 5M : 5 En : 4 iz ¢ 20 M os: me att 5 35 M 5 *e. ir \ A 2 y 5M. =~ 20 M. 35 M Proceedings Royal Acad Amsterdam. Vol. IX, H. WIND. „Current-Measurements at various depths in the North Sen,” CURRENTS OBSERVED at H, (Lat. 53°44’, Long. 4°28’). 1905, November 7—8. + c . . . e . ee ¢ e a Aaa e % . . ee de . ni . . re eee ce ON. (San 6/ . 4 i= —— ee . 4 ) 0 (Truc diveetion.) 1906, Februar 7—8. CENTRAL DIAGRAMS. em Se Te BG % miles p. hour. PROGRESSIVE DIAGRAMS. 1906, May 2—3. . . . wp aA We el . oe . . Ge . ” a4 . . „ . . nn ed Averages (at figures indicate moon-hours), sas x x es, \ tje ja Se, (573 ) , that the deviations for a part considerably exceed the limits of accu- rateness of the statements. It should be observed that the charts refer to currents near the surface, whereas the values of the table derived from our observations refer to a depth of 5 M. Finally we may mention that the observations at station H2 up till now have been continued in the same way, that is to say, they are still made every quarter of a year, as far as possible, during 24 hours. Moreover, owing to the kind co-operation of His Excellency the Minister of Marine, a current-meter of PeTTERSSON has been placed on the lightship ‘Noord-Hinder’, with which since November 1906 daily, in so far as the state of the weather permits, with intervals of three hours, measurements at various depths are made by the ordinary staff of the lightship. The lists of observation are forwarded to the ‘“Rijksinstituut’? and promise to yield important material, especially for the inquiry into the way in which the tidal and residual currents differ in layers of different depth. Mathematics. — “The locus of the pairs of common points of n+1 pencils of (n—1)-dimensional varieties in a space of n dimensions.” By Dr. F. Scnun. (Communicated by Prof. P. H. Scnoure). edet (4) ¢—1,2,;..,%-+-1) be Jed. pencils of. (n— 1)- dimensional varieties in the space of operation Sp” of n dimensions and let 7; be the order of the varieties V;of the pencil (V7;). Let moreover a; be the number of points of intersection of the » varieties eee ie Sa ey, Vies «+ -5 Vase not’ of, necessity lying in the base-varieties. When considering the locus of pairs of points 2, 2?’ through which a variety of each of the pencils passes we have exclusively such pairs in view of which neither of the two points lies of necessity on a base-variety of one of the pencils and we call the locus thus arrived at the locus proper L. We determine the order of Z, out of its points of intersection with an arbitrary right line /. To this end we take on /an arbitrary point Qio..., and we bring though it varieties V,,V,,V,,..,Vn, having an41—1 points of intersection not lying on Qje..., and the base- varieties. Through each of those points we bring a V,,41 and arrive in this way at a,4;—1 varieties V,41 intersecting together line / in (An4-1—1) rn points Qu. So to Qian correspond (dn41—1)7'n41 points Q,41. To find reversely how many points Qi. „ correspond to Q,+1 we take arbitrarily on / the points Q;41, Qi42, Qi43,..., Q,+1 and we bring through those points respectively a V;41,Vi40,Vi+s,.--, Viti. We now put the question how many points Qis3.,.; lie on / in such a way that the varieties mentioned V;41, Vi+e,..,Vn+1 and the varieties V,,V,,..,V; passing through Qj)o3...; have a com- mon point not lying on the base-varieties. For 7 <{x the answer is: ar, Har, +... Hair. To prove this we begin by noticing that the correctness is imme- diately evident for = 1. If we now assume the correctness for ¢ = j, we have only to show that the formula also holds for 7=)-+ 1. Given the points Qj42, Qj43,.--, Cui. To determine the number of points Qio3...;41 we take on / an arbitrary point Qi23...;, we — V.V; and the varieties V;40, Vj;43,..., Va41 resp. passing through Q)+42,Qj;+3,.-.,Qr41 we bring a variety V ;41; these aj 41 varieties V;4, cut / in Aja 7741 points Q;4 1. So to Qios... ; corre- spond aj417;41 points Q)4 and (according to the supposition that the formula holds for 7=J) reversely to Q;41 correspond ar, Har, + +...+a,r; points Qios...;- So there are ar, + a,r, +... + + Oe a Ai Tj coincidences OEE er these are the points Qiz...j jar belonging to the given points Q;+2, Qj1s,..-, Qn1s in this way the correctness of the formula has been indicated for When asking after the number of points Qjs..., corresponding to Qi we have i=n, so that the formula furnishes a,7,-+ a,r, + +....+a,r, for it. This number must however still be diminished by 71, as each of the points of intersection of / with the W+, passing through Q,.1 is a point of coincidence Qjs3....,—1 Q, but not one of the indicated points Qi... So on / there exists between the points Qs...» and Q,4 1 an a rn ee + An — Pii) Correspondence. The ar, Har, +... A Qa+17+41— 27,41 coimeidences are the points of intersection of / with the locus Z to be found and the points of intersection of / with the (n— 1)-dimensional variety of contact RVie...» of the pencils (V), (V),-..,(V‚); we understand by that variety of contact the locus of the points, where the varieties VW, V,,.--, Vn passing through them have a common tangent, so where the (2 —1)-dimensional tangential spaces of those varieties cut each other according to a line. ( 575 ) 2. To determine the order of R Vie. „ we must observe that RVi, is the locus of the points of contact of the varieties V,, with the eurves of intersection Ci2.. »—1 of the varieties V,, V,,...., Vui. So the question has been reduced to that of the order of the variety of contact of a system of oo! (£ —1)-dimensional varieties and a system of oo”—! curves. That order can be determined out of the points: of intersection with an arbitrary line /. In a point of intersection of / with a variety of the system we bring the (2 — 1)-dimensional tangential space Sp"! and in a point of intersection of / with a curve of the system the o”—? tangential spaces Sp"—'. If we act in the same way with all varieties and curves of both systems, then the tangential spaces of the varieties furnish an 1-dimensional envelope HE, (i.e. a curve) of class u + v (as is evident out of its osculating spaces Sp"—! through an arbitrary. point of /) with v osculating spaces Sp"—' passing through 1; here u is the number of varieties of the system passing through an arbitrary point, and » that of the varieties touching an arbitrary right line. The tangential spaces of the curves in the points of intersection with / have an (n — 1)-dimensional envelope KE, of class p + w with l as w-fola line, where g is the number of curves of the system passing through an arbitrary point and wp that of the curves touching an arbitrary space Sp"—! in a point of a given right line of that space; for, if we bring through a point Q of / an arbitrary Sp*—2, then each of the p curves of the system passing through Q furnishes a tangential space Sp”—' passing through this Sp"—? whilst the space Sp"—! determined by / and Sp"? (just as every other Sp”—! passing through /) is w times tangential space of the envelope. Both envelopes have thus (u + rv) (p + wW) common tangential spaces Sp"—'. Each of the r osculating spaces Sp"—! of /, passing through / is a w-fold tangential space of #,, so it counts for wp common tangential spaces; so that up + uy + ry common tangential spaces not passing through / are left; these indicate by their points of intersection with / the points of intersection of 7 with the variety of contact, so we find: The (n—1)-dimensional variety of contact of an o* system of (n—1)-dimensional varieties of which u pass through a given point and v touch a given right line, and an oor—! system of curves of which p pass through a given point and w touch a given space Sp’! in a point of a given right line of that space, is of order LW + PP + UG. 3. With the aid of this result it is easy to determine the order ( 576 ) of the variety of contact (locus of the points with common tangent) of n simple infinite systems (u, ¥,), (Uy: Pals. + +++ (ny Pn) Of (n—1)- dimensional varieties. This order is Pile Ps, Dn Us eee rl —f—+...+—4+n—-1], u, Un 2 as can be shown by complete induction. The formula holds form = 2. We assume the correctness of the formula for n=: and out of this we must find the correctness for n =z - 1. The variety of contact for #41 systems in Spit! is the variety of contact of the system of varieties (f,, »,) and the system of curves formed by the intersections of the 7 remaining systems of varieties. So we have: MS, PSM» PH My Uy- ijt The points of contact of the curves of the system with a given space Sp’ form the (¢—1)-dimensional variety of contact of the sections of Sp’ with the systems (u,, v.), (Us, P's)» « ++ « » (Mi+1, Pipi); these sections are likewise systems (u,v), « « « « 5 (4i+1, Pii), but of (¢—1)-dimen- sional varieties. The variety of contact mentioned is according to supposition of order tess ( EEE + Edi). Mar is Mi+-1 The points of intersection of that variety of contact with a right line / of Sp’ being the points of / in which Sp’ is touched by curves of the system, we have: vp Y Pv; Patent (Ee de Hjir). U, Us Witt Thus according to the formula wb + rp + up the order of the i-dimensional variety of contact of the 7-+ 1 systems of varieties becomes YP yv Vv; 1 : mone wen (ett), u, u, Wii by which the correctness of the same formula for n=? 1 has been demonstrated. So we find: For n op* systems (pus?) (far Pa) ++ += 9 (Uns Pn) Of (n—d)-dimen- sional varieties the locus of the points where the varieties of the systems passing through it have a common tangent rs an (n—1)-dimen- sional variety (variety of cantact) of order 1 De AEN Vn HE ho. 3 tly en ee gl w, b Un 4, ( 577 ) If the systems are pencils, then Bea LON pee 2 (rl); thus the order of the variety of contact RV...» is: 2(r7,,t+7+..-+tm)—n—1. 4. Returning to the correspondence between the points Qis.,,,and Q,41 we find for the number of coincidences which are points of intersection of / with the demanded locus £, i. e. for the order of L: ete Oy hy ee af On ee en ie =e = te Mei) i=n+l1 +n+1= 2 (a: — 2)7,4 1h. i=] It is easy to see that a base-variety B; of the pencil (V;) is an (a; —1)-fold variety of L. The tangential spaces Sp"—! of L in a point P of B; are the tangential spaces in P of the varieties V;, which are laid successively through one of the a—1 points of intersection (not lying on P and the base-varieties) of the varieties ee ees een Mrne Vr Dassmie through: P: So we find: Given n+1 pencils (V;)(¢=1,2,...,n +1) of (n —1)-dimen- stonal varieties in the space of operation Sp”. Let r; be the order of the varieties of the pencil (Vi) and a; the number of the points of imtersection (not lying on the base-varieties) of arbitrary varieties eee Pen clis UV CV (Vr | Mie ye Vagal Le locus proper of the pairs of points lying on varieties of each of the pencils is an (n — 1)-dimensional variety of order | ital! = (as: — 2) ri; + 1}, Jl having the (n — 2)-dimensional base-variety of pencil (V;) as (a—1)- fold variety. If np >>3, then also in the general case the base-varieties of the different pencils will intersect each other. In like manner as we have dealt with pencils of surfaces *) we can also determine the multiplicity of common points, curves ete. of base-varieties. Sneek, Jan. 1907. 1) See page 555. 39 Proceedings Royal Acad. Amsterdam. Vol. IX, (578 ) Astromony. — “On the astronomical refractions corresponding to distribution of the temperature in the atmosphere derived from balloon ascents.’ Preliminary paper by H. G. vAN DE SANDE BAKHUYZEN. 1. The various theories of the astronomical refraction in our atmosphere consider the atmosphere as composed of an infinite number of concentric spherical strata, each of uniform density, whose centre is the centre of the earth and whose densities or temperatures and refractive powers vary in a definite way. The various relations between the temperature of the air and the height above the surface of the earth, assumed in the existing theories, are chosen so, that 1*t they do not deviate too far from the suppo- sitions on the distribution of the temperature in our atmosphere, made at the time when the theory was established, 2rd that the formula derived from this relation for the refraction in an infini- tesimal thin layer at any altitude could be easily integrated. At the time when the various theories were developed, only little was known about the variations of the temperature for increasing heights, and this little was derived from the results of a small number of balloon ascents and from the observations at a few mountain- stations. In the last decade, however, ascents of manned as well as of unmanned balloons with self-registering instruments have greatly increased in number, and our knowledge of the distribution of the atmospheric temperature has widened considerably, and has become much more accurate. Now [ wish to investigate, whether by means of the data obtained, we can derive a better theory of refraction, or if it will be possible to correct the results of the existing theories. ae been derived from the following publications : I. Ergebnisse der Arbeiten am aéronautischen Observatorium Tew 1900—1902, Band I, I and III. Il. Travaux de la station Franco-scandinave de sondages aériens a Halde par Teisserene de Bord. 1902—1903. Ill. Veröffentlichungen der internationalen Kommission fiir wissen- schaftliche Luftschifffahrt. From the last work I have only used the observations from December 1900 till the end of 1903. I wished to investigate the distribution of the temperature up. to ihe greatest heights, ati therefore 1 used for my researches only the balloon ascents which reached at least an elevation of 5000 meters; 2. The temperatures in our atmosphere at different heights have and, following Hereeseur’s advice, I have used only the temperatures observed during the ascents, as during the descents aqueous vapour may condense on the instruments. It is evident that for the determination of the refraction, as a cor- -rection to the results of the astronomical observations, we must know the variations of the temperature at different heights with a clear sky. For the temperatures, especially of the layers nearest to the surface of the earth, will not be the same with cloudy and uncloudy weather, as in the first case the radiation of the earth will lower the temperature of those layers, and so cause an abnormal distribution of temperature. It is even possible that in the lower strata the temperature rises with increasing height, instead of lowering, as is usual. For this reason I have divided the balloon ascents into two groups, Ist those with a cloudy sky, 2rd those with a clear or a partly clouded sky. . In working out the observations, I have supposed that for each successive kilometer’s height the temperature varies proportionally to the height, and after the example of meteorologists, I have deter- mined the changes of temperature from kilometer to kilometer. For this purpose, I have selected from the observations, made during each ascent, the temperature-readings on those heights, which corresponded as nearly as possible with a round number of kilometers, and I have derived the variations of temperature per kilometer through division. The available differences of height were often less than a kilo- meter, especially at the greatest elevations; in those cases I adopted for the weight of the gradient a number proportional to the difference of heights. Sometimes on the same day, at short, intervals several ascents have been made at the same station, or at neighbouring stations, from which the variations of temperature at the same heights could be deduced. In these cases I have used the mean of the results obtained, but I assumed for that mean result the same weight as for a single observation, as the deviations of the daily results from the normal distribution of temperature are only for a small part due to the instrumental errors, and for the greater part to meteorological influences. 3. The observations which I have used, were the following: from publication-I, 31 ascents of which 12 had been made in pairs on the same day, so that 25 results were obtained ; from publication HI, 88 ascents all on different days; and from publication HI, 170 39* ( 580 ) ascents distributed over 119 different days; — 1 have disregarded the observations. marked as uncertain in this work. On the whole I have obtained the results on 182 different days, of which 58 with unelouded and 124 with clouded sky. The temperature gradients for each month were derived from this material, and to obtain a greater precision, I have combined them in four groups, each of three successive months, December, January and February (winter), March, April and May, (spring), June, July and August, (summer), September, October, November, (autumn). BA B ABk Variations of temperatures per kilometer. (V.T. Variation of temperature per kilometer; N. Number of observations). A. Clear sky. | Winter. | Spring. | Summer. | Autumn | Mean. Kil Vas NN | NEE N N. | Vel, | N. oil fo} 10 | Se ws |l—2s| is |H oe 15 Co se 1-24 10 54 15 | 4.3) 18 |-3 15 | a 58 9— 3 ||/- 5.2) 10 |i— 4.9) 15 |i— 4.4 18 | 4.6) 15 |I sl 58 3— 4}1— 5.4) 10 I= 5.8 15 ||—- 5.4" 18 ||— 5.3) 15 ||/— 5.5; 58 5 15.3) | 6.7 1438 9 18 57 149 | 5.9) 572 5— 16 i— 5.6) 89 pare | 13.6 |I— 6.0| 18 ag 138 |I— 6.5) 543 6— 7 ||— 5.8] 8 [|= 7.5} 127 ||— 6.6) 173 ||— 6.7| 10.1 ||— 6.7) 48.1 7— 8 ||— 6.8} 7 |i— 7.8 108 |— 7.5} 146 |— 8.0} 8 ||— 7.5) 404 8— 9 ||I— 7.6 5 |j— 6.4 78 - 7.4, 13.3 |J/— 8.1) 8 7.3] 34.1 9—10 ||— 5.9} 4 || 4.4) 5.7 |-—- 7.2] 13 | — 6.9) 7 | 6.4| 29.7 10—11 ||— 3.8} 29 ||- 2.5) 5 ||— 6.8 104 ||— 6.4] 68 |— 5.4) 251 11—12 || 6 2} 2 |]— 2.4) 26 ||— 5.9) 52 IJ 2.0) 59 |I 3.5) 15.7 1243 || 1.6) -2 |H 2.0) 1 —1.4; 2 |/—1.0 49 — 0.7) 9.9 13 —44 + 7.0) 1° [HH4.0 2 | 4.0) 16 | 0.8) 46 14—15 + 0.7 16 ||— 5.4] 1 | 41.5) 26 15—16 TOE SOB 4 ( 581 ) B. Cloudy sky. | Winter. | Spring. Summer. | Autumn. | Mean. | | ns CAERE Ii A0 od es Ne Le WE | Na Uvarov UNS ETL PNG GEen oe io eel aa lect 4 Fol 40 Pel 14 4— 2 3.0) 27 || 5.6) 325 |— 5.4) 24 ||— 3.7) 49 ||— 4.3) 1235 Q9— 3 ||— 4.5) 27 | 4.8) 33 | 5.4] 24 | 4.3) 40 | 4.5] 124 3— 3 | 5.8) 27 |} 5.5) B | 5.4) 238 || 5.8) 395 |— 5.6) 1233 s— 5] 6.8] zi || 6.7) 33 |- 6.4] 23 ll_64| 99 ll 6.4 122 5— 6 ||— 6.9} 26 |— 6.7] 30.7 || 6.7] 215 ||-- 6.2] 365 || - 6.6] 114.7 G— 7 || 6.8) 254 |— 6.7) B | 6.6) 177 | 7.3| 278 ||— 6.9] 959 7— 8 |- 6.9] 197 |— 7.2) 203 || 7.9] 168 || 5.9] 216 |—- 6.8] 784 SS Ga (ANO fea ="7 9) 147 ||— 7:9) 13 |= 6.9) 515 9-0 |=: 6.2). 12.3 I= 3.9) 129 | 8.4) 121 I= 7.5] 11.4 | 6.5) 4847 10—11 | 5.4) 94 | 1.8 96 ||— 5.9 81 |I- 5.4 85 || 4.5] 356 44—12 ||— 2.5) 76 4.0 83 || 2.4) 51 1.9 68 |L-- 1.2) 278 42—13 |j— 1.3) 5 |H 1.2 67 |H 0.2) 19 | 0.5) 41 HHOA) 177 1314 ||— 0.9] 27 |I-3.9| 1 aa rde IE OKBK BAI 1415 |H 1.9 19 32 1 + 0.2} 29 1516 [— 0.6 1 |— 3.2 05 =d 15 16—17 || 0.4] 08 | + 0.1) 08 We may derive from these tables that the mean variation of temperature with clear and with cloudy weather only differs in the lower strata, but is nearly the same in the higher ones. In order to deduce from the numbers in this table the temperatures themselves from kilometer to kilometer, I have also derived from the data the following mean temperatures at the surface of the earth: clouded sky clear sky Winter + 0°1 — 0°.9 Spring + 6 4 + 5.1 Summer +14 4 + 14.7 Autumn + 9.0 + 7.9 By means of these initial temperatures and the gradients of table I ( 582 ) C. Cloudy and uncloudy sky. | Winter. | Spring. | Summer. | Autumn. | Mean. | | | Kil. | LEE N. |Hann.j| V.T. | N. haan V.T. | N. | Hann.|] V.T.| N. |Hann.f| V.T. | N. o—4|-Polsr | 405[-2s}4s | Sal -Sale | Lalas | Srei | 1— 2|-3.3/37 | —2.9/|-5.6 415| —5.5]-4.7] 42. | —5.6||—3.6] 55 | —4.1||—4.3) 1815 |—4.4 2— 3|—4.7/37 | —5.0)—4.9) 48 | —5.4]|-4.8)42 | 54.455 | ASAT 182 |—5.0° 3— 4/|-5.7/37 | —5.8|-5.6/48 | —5.8 5.9] 418 | —5.6||—5.6| 545} —5.8/—5.5| 181.3|—5. = lade 37 | —6.7||-6.7/ 413| —6.7 6.041 | 5.96.0 539 mens 179,2 —6.3 5— G|—6.7/349 | —6.7]|-6.8| 443] —7.3 64305) —6.4ll_6.5,503| 6.86.6] 169 1-68 6— 76.6) 33.4 | —6.7|| -7.0|37.7| —7.9]|-6.6)35 | —7.9||-7.4] 379] —7.4]|—6 9144 —7.0 7— 8|—6.9| 26.7) —7.9||—7.4/31.1| —6.3]|—7.4) 31.4} —7.7||-6.4| 296| —7.3]|--7.3] 1188 |—7.4 8— Y}|—6.5) 19.2 | —6.9)|—6.1/24 —6.4 7.6 214. —7 .6||—7.9} 21 —7.6||—7 1| 91.6 |—7.4 940 6.9) 163, —6.4]|4.0 186 —4.8)—7.8 25.1| —6.9]|-7.4| 184| —6.6—6.5| 78.4|—6 10—11}| 5.0) 12.3) —3.9|—2.0/ 14.6; —0.9|| 6.4) 185} —5.0]|-5.7| 15.3] —6.1]|—4.9) 60.7|4.0 1119 —2.6| 96| 0.0 ||-0.2} 10. | +0.5]}—4.0| 103 —9.4| 4.9] 127] —2. 2.1) 435|/—1.2 1913 43 7 | 44.3 a1| 0.5) 39 —0.5| 9 —0.2 27.6. 43—14]|—0 9) 2.7) 4.6/2 41.0} 2 4°31 3 EE a 14—15)44.9| 19 Sal | 40.7| 16) EB wel _06l 55] 15—16|—0.6, 1 | —3. 05 0.8 1 —0.6) 25 1617/01 08 | | | +04} 08 as ar Pl | which in a few cases have been slightly altered, I have derived the following list of temperatures for clear weather from kilometer to kilometer. Although the adopted values for the temperature of the air above 13 kilometer are not very certain, yet the observations indicate that at these heights the temperature decreases slowly with increasing height. The refraction in those higher strata being only a small part of the computed refraction, nearly */,,, an error in the adopted distribution of temperature will have only a slight influence on my results. 1 must remark that almost all the observations have been made during the day, generally in the morning. It is evident that the varia- tion of temperature, especially near the surface of the earth, is not the same during the day and during the night, but the number of ( 583 ) MADB, GB 1. Temperatures at heights from 0 to 16 kilometer for clear weather. Winter. | __ Spring. | Summer. | Autumn. | Mean. Height. |Temp.| Diff. |Temp.| Diff. |Temp.| Diff. |/Temp.| Diff. Temp. Diff. 0 1°9 + 54 444.7| 17.9 + 6.4 EZ „Jl DS AES iks >. 4! 7 1.2 BER —3.6 3 —2.8 +0.6 —1.1 — 0.7 a deb. 8 5 5.3) 5 , a —4.2 —5.4 a —4,3 ef —3.2 jn —4.3 — 4. — 3.9 76 5.3 1.0 F —5.2 —4.9 a —4.4 aM —4.6 5 —4 8 : —10.4 — 8.8 3.2 0.7 — 3.8 —5.4 —5.8 rr —5.4 Ti —5.6 —5.5 4 |—15.5 —14.6 — 2.2 — 4.9 — 9.3 —5.8 — 6.7 —5.9 —6.1 —6 1 5 |—21.3 —21.3 — 81 —11.0 —15.4 —6.0 —6.7 —6.0 —6.9 "—6 4 6 |—27.3 —28 .0) —14.1 —17.9 —21.8 —6.2 —6.9 | —6.6 —7.2 —6.7 7 —33 5 —34.9 - 20.7 — 25.4 —28.5 —6.8 —7 3 —7.3 —7.7 —7.3 8 |—40.3 —42 2 —28.0 —32.8 —35.8 —7.3 —6.9 —7.6 —7.6 —7.4 9 |—47.6 —49 A —35.6 40.4 -43.2 —b6.4 —5.4 —7.2 —b6.9 —6.4 10 |—54.0 —54 5 —42 8 —A7 3 —A49 6 —4.9 —25 | —6.8 —6.1 —5.1 41 |—58.9 —d7 .0 —49 6, —53.4 —54.7 —2.4 | —1 0 —4.0 —2 0 - 2.3 12 |—61.0 —58.0 —53.6 —5d.4 —57.0 —1.0 —1.0 —1.0 —1.0 —1.0 13 |—62.0 —59.0 —54.6) —56 4 —58 0 —0.6 —0.6 —0.6 —0.6 —0.6 14 |—62.6 —59 6 —55.2. —57.0 —58.6 —0.4 —0.4 —0.4 —0.4 —0.4 15 |—63.0 —60.0) —55 .6) —57.4 —59.0 —0.2 | —0.2 | —0.2 —0.2 —0.2 16 |—63 2 —60.2 —55.8| —57 .6 —59.2 | | observations was not great enough for a reliable determination of this difference. Lastly I remark that the various balloon ascents have been made from different stations, Halde (in Danemark), Berlin, Paris, Strasbourg and Vienna and consequently the given values do not hold for one definite place, but for the mean of the area enclosed by those stations. After I had derived the temperatures given in table II, I got notice of two papers, treating of about the same subject, namely: J. Hann, Ueber die Temperaturabnahme mit der Hohe bis zu 10 Km. nach den Ergebnissen der internationalen Ballonaufstiege. Sitzungsberichte der mathematisch-naturwissenschaftlichen Klasse der K. kK. Akademie der Wissenschaften Wien. Band 93, Abth. Ila, S. 571; and S. ( 584 ) GRENANDER. Les gradients verticaux de la température dans les mmima et les maxima barometriques. Arkiv for Matematik, Astronomi och Fysik. Band 2. Hefte 1—2 Upsala, Stockholm. Of the results which Hann has given, up to a height of 12 kil., I have taken the means of groups of 3 months, which are printed in table I by the side of the values [ had obtained; the agreement of the two results, which for the greater part have been deduced from different observations, is very satisfactory. GRENANDER in his paper chiefly considers the relation between the changes of temperature and the barometer readings; his results cannot therefore be compared with mine directly, but probably we are most justified in comparing the variations of temperature at barometer maxima, with those which I have computed for clear weather. For great elevations, till nearly 16 kil., GReNANDER also obtains with increasing height a small decrease of temperature. It is difficult to state with what degree of precision the tempe- ratures of table IL represent the mean values for the different seasons; the deviations, especially at great heights, may perhaps amount to some degrees, but certainly they represent the mean distribution of temperature better than the values adopted in the various theories _of refraction, and we can therefore derive from them more accurate values for the refraction. 4. It is hardly possible to represent the relation between the temperatures in table II and the heights by a simple formula, and to form a differential equation between the refraction, the zenith distance and the density of the atmosphere at a given height, which can be easily integrated. Therefore I have followed another method to determine the refrac- tion corresponding to the distribution of temperature I had assumed. According to Rapav’s notations (Essai sur les refractions astrono- miques. Annales de l’Observatoire de Paris. Mémoires Tome XIX), the differential equation of the refraction, neglecting small quantities, is : l (: — ay —3 co)) dw t? 9 f, ) 0 2 5 Fa) i 2-2 y — ly cot? z + 2 p(y — ew Rp! R (I) ds = a Here is: R radius of the earth for 45° latitude, yr, radius of the earth for a given point, h height above the surface of the earth, ( 585 ) =r, +h, u, index of refraction at the surface of the earth, {4 DE) EE) ” ” > height h, 9, density of the air at the surface of the earth, e density at the height A, ¢, temperature at the surface of the earth, /, height of a column of air of uniform density at 45° latitude, of a temperature ¢,, which will be in equilibrium with the pressure of one atmosphere, the gravity being the same at different heights. According to RecGnavunt’s constants, we have /, = 7993 (1 + at) meter, if @ represents the coefficient of expansion of the air. Between these quantities exist the following relations: 3 : o u =l + 2eo (ec being a constant), w=l—— 9, CO, ? a hk Ith C= == U aa, EE =d RTT EEEN : 1+ 2co, sin 1 be (7, +A), To determine the value of ds at each height, we require a relation between w and y or between w and h, which can be obtained when we assume that the temperature varies according to Ivory’s theory, or that the temperature varies as represented in table II. For the same given values of z and w, the two values of ds in formula (I) can be computed by means of the first and by means of the second supposition, and the differences of these two values of ds can be found. By means of mechanical quadrature, we can then determine the differences As of the refractions s according to Ivory’s theory and according to table IL. The relations between y and w may be found in the following manner. 5. If in a given horizontal initial plane, at a distance 7, from the centre of the earth, the pressure is p,, the temperature f, and the density of the air @,, and in another horizontal plane, / kil. above the former, the pressure is p, the temperature ¢, the distance from the centre of the earth 7, and the density of the air e, then we have (see RADAU) : R R Rh r Rh a(2)=—2 (2) na — trad or mal E)=-lal). Po Pa” Qo To 5 R \p Qo lr ting © bee or, putting — = % anc =a i: 0, (r, +h) l, be fal BY fie Waste dap. oe WAM EEE CLT R jz es Te ( 586 ) further is: l t 1 (Lt —t 8 en n=(1—9)y, - . UD pel eae 1-+at, a An lta When dividing equation (ID) by (ID, we get: 5) d{| — Te AD dy R p Le 1-9 if we put ? From the two last equations follows: : 1 pase jan hie Rk n =% jaa + 0-9) oe According to Ivory’s theory ® = fw, where f is a constant value (RapaAu assumes 0,2); if we introduce this relation into the equation (LI) we obtain after integration : y = 0,40 — 1,8420681 © Br. log (1—w) REL. By substituting (V) in (1) we can therefore calculate for each value of w the value of ds according to Ivory’s theory. 6. Now I proceed to determine the relation between w and y according to the temperature table II. Of two horizontal planes, one above the other, the first is situated n kil. (2 a whole number), the second n’ kil. (v’ = or << n+ 1) above the surface of the earth; their distances from the centre of the earth are 7, and 7,, their temperatures ¢, and ¢, and the values of y, y, and yv. The temperature between n and n’ varies regularly with the height and, to simplify the formulae, I suppose ¢, — t,’ a (ent) Tik k a Dn (uw rn Yn) — Cn dD, . . . . ° e . = (V I) proportional to y,— Yy,, so that, if 9, = R Hence follows dye, d? and after substitution of dy in (IV) Tn ( 587 ) and integration (on a 1) lg (1 Ek D,) — ly (1 = w) ° 5 5 - (VII) in which 1—o represents the ratio of the densities in the two horizontal planes. If we substitute n +1 for n', we can find in table IL the tem- perature for the two planes and hence also #,; as y, and y,4) are also known, we can derive from (VI) the value of ¢, and we can deduce from (VII) the ratio of the densities in those planes. By put- ting for ” successively O, 1, 2, ete. we can construct a table con- taining the densities of the air, D,, D,, D, ete. at the height of 1, 2, 3, etc. kil. above the surface of the earth, the density at the surface being unity. © It is easy to derive from this table the height of a layer of a given density d. If d< D, and > D, 41, the layer must be situated between n and n+ 1 kil., and we oniy want to know in which manner, within this kil., the density varies with the height / above the lower plane. We may assume: ts ns Di Bo ek di Diders vhenee a=— lq ze n a being known, we may determine for each value of d,/ and also y. By substitution in (I) we find then for each value of w the value of ds. 7. Now we are able to form the differences of ds after the theory of Ivory and after the table of temperatures LU, for values of w which increase with equal amounts, and then determine the whole diffe- rence of the refraction for both cases. For great values of z and small values of y and w the coefficients of dw in (I) will become rather large, which derogates from the precision of the results. This will also be the case when the differences of the successive values of w are large; small differences are therefore to be preferred, but they render the computation longer. . Both these difficulties can be partly avoided if, according to Rapav’s remark, we introduce (/w as a variable quantity instead of @ ; the value of ds thus becomes: ( 588 ) (i — —(y — seo) Yow Ve oR ss: +y I a eee fh or approximately : is = dYw pit V/a: te 7 } 0 It is evident that for small values of © the coefficient of do in (VIII) is smaller than that of dw in (1), and that also the refraction in the lower strata will be found more accurately by means of the formula (VIII) than by means of (I). For if we increase V@ in formula (VIII) and @ in formula (I) with equal quantities, beginning with zero, we find, that, from w —=0 to w= 0,2, the number of values in the first case is twice as great as in the second case, hence the integration by means of quadrature will give more accurate results in the first case. Therefore I have used the formula (VIII) and computed the coef- ficient of dV for values of Ww, 0, 0,05, 0,10,0,15... to 0,95. The density of the air which corresponds to (/w = 0,95, occurs at the height of about 18 kil. From the observations at my disposal I could not deduce reliable values for the temperature at heights above 16 kil.; yet it is probable that the gradients at those heights are small and I have assumed the temperature at heights of 17 and 18 kil. to be equal to that at a height of 16 kilometers. In this way I have determined by means of mechanic quadrature, and an approximate computation of the refraction between Vw = 0.925 and Vw == 0.95, the differences As of the two values of the refraction corresponding to Ivory’s theory and corresponding to the table of temperature II in the part of the atmosphere between the earth’s surface and a layer at a height of about 18 kil. where fw is 0.95. I have worked out this comptuation for the zenith distances 85°, 86°, 87°, 88°, 88°30', 89°, 89°20’, 89°40’ and 90°. An investigation, made for the purpose, showed me that in for- lo mula (VIII) the terms = (y — 3ew) in the numerator in the denominator may be neglected for all zenith distances except — 90°; therefore I have taken them into account in the computation of the horizontal refraction only. The results which I have obtained for the differences : ( 589 ) As = Ivory—table of temperatures are the following: ie te Bie Be II. Refraction after Ivory — Refractions after the table of temperatures 11. ze RY ee) cee nn |: wea yee zeen Winter | Spring Summer Autumn ne Ta pee ne si is. | ‚Winter | Spring | Summer | Autumn 350 =. ONDA | + 0"78) + 0"66| + 0”31 + 0"49 | + 0/28 | — 029 | — 0/17 | 4 0/18 86° + 0.13 | + 1.26) + 0.95) + 0.30 | + 0.66 | + 0 53 | — 0.60 | — 0.29 | + 0.36 87° = 0.47 | + 2.08) 4+ 1.341) — 0.20 | + 0.66 | + 1 13 | — 1.42 | — 0.65 | + 0.86 88° —3 93 + 3.10) + 0.95) — 3.29 | — 0.83 | + 3 10 | — 3.93 | — 1.78 | + 2.46 88930! | — 9.64 | + 3.06) — 0.67| — 8.51 | — 3.95 | + 5.69 | — 7.01 | — 3.28 | + 4.56 89° —23.69 | + 1.08) — 5.45) —21.15 | —12.34 | 441.38 | —13.39 | — 6.86 | + 8.84 89°20' | —43.80 | — 3.17) —12 68) —38.77 | —24.51 | 4419.29 | —21 34 | —11.83 | 444.96 89°40' |—1' 2497, —13.07) —25.25|—1'11'16, —27.74 "134.93 —34.67 | —22.49 | +23.42 90° —2 32.4 | —33.1 | —52.9 |—2 9.6 | —1'30"9) +1'1"5 | —57.8 | —38.0 | H38.7 To test the computations, we may compare the mean of the values of As for the four seasons, and the values of As in column 6 which have been computed, independently the former, for the mean yearly temperatures, which are almost equal to the mean of the temperatures in the four seasons. Only for z= 89°40' and z= 90° do these values show deviations exceeding 0".1. From table II follows, 1 that for a distribution of temperature, as derived by me from observations, the refraction deviates percep- tibly from that deduced from Ivory’s theory, 2 that the differences in the refraction in the different seasons are about of the same order as the deviations themselves. I want it to be distinctly understood, 1 that the adopted distribution of temperature above 13 kil. and especially from 16 to 18 kil. is uncertain, and 2 that I have not taken into account the refraction in the layers which are lying more than 18 kil. above the surface of the earth, in other words those layers where the density, as compared to that of the surface of the earth, is less than 1 — 0,957, or less than 0,0975. (590 ) Physiology. — “An investigation on the quantitative relation be- tween vagus stimulation and cardiac action, on account of an experimental investigation of Mr. P. WorrersoN” *). By Prof. H. ZWAARDEMAKER. (Communicated in the meeting of December 29, 1906). The experiments were performed on Emys orbicularis, whose right nervus vagus was stimulated by means of condensator charges and non-polarising electrodes of Dorpers *), while auricle and ventricle were recorded by the suspension method. The mica-condensators had a capacity of 0,02, 0,2 and 1 microfarad, the voltage varied from a fraction of a volt to 12 volts, occasionally even more. From this the intensity of the stimulus was calculated in ergs (or in coulombs by Hoorwxe’s method). Only a part of this energy, passing through the nerve, when it is charged, acts as a stimulus. What part this is remains unknown, but it is supposed not to vary too much in the same set of experiments. In the typical experiments a summation took place of ten stimuli, succeeding each other in tempos of */, second; in particular experiments single stimuli or other summations were investigated. Of fatigue little evidence is found with our mode of experimenting, rather a somewhat increased sensitiveness of the vagus system towards the end of a set of experiments. Stimulation of the right vagus produces in the tortoise in the first place lengthening of the duration of a cardiac period *), in such a way that in the second period, after a stimulus, starting during the cardiae pause, the diastolic half of the period is considerably retarded, while in some subsequent periods a decreasing retardation of the diastolic part of the period is noticed. Then stimulation of the vagus causes contraction to become feebler, this phenomenon becoming gradually more distinct and reaching its maximum some periods after stimulation. This decrease of contractile power is primary, since it may also occur when any change in the automatic action is absent (e.g. when the stimulus consists of one condensator charge and when the left vagus is stimulated). Finally vagus stimulation as a rule produces slackening of the tonus, rarely tonie heightening. Changes in conductivity were only observed once. 1) For details we refer to the author’s academical thesis, which will be published ere long. 2) Onderzoekingen Phys. Lab. Utrecht (3) Vol. I p. 4, Pl. I, fig. 1, 1872. 3) The duration of a cardiac period is reckoned from the foot-point of a sinusal contraction or if this is not visible, of an auricular contraction, to the foot-point of the next following sinusal resp. auricular contraction. ( 591 ) The negative chronotropy holds good for sinus, auricle and ven- tricle to the same extent, the negative inotropy exists exclusively for the sinus and the auricle, is mostly positive for the ventricle, if it is found; the tonotropy is met with in auricle and ventricle. A latent stage of the phenomenon, measured by the time-difference between vagus stimulation and vagus action, was always observed. It is smallest for the inotropy; already the first period often shows an enfeeblement of the contraction, which in the subsequent periods increases still further. The latent stage of the chronotropy is greater, for only in the second, sometimes in the third period, a retardation is noticeable; on the other hand this phenomenon reaches its maximum at once. Inotropy and tonotropy do not coincide. On the contrary, the maxima of effect form the following series as to time: first maximum of chronotropy, then maximum of tonotropy, finally maximum of inotropy. In regard to the sensitiveness for vagus stimuli, we remark that for the inotropy the “threshold value” lies below that for the chrono- tropy and for this latter lower again than for the tonotropy. So we have: Threshold value for inotropy < idem for chronotropy < idem for tonotropy. From the fact that dromotropy did not occur in our experiments, one would infer that the threshold value of the dromotropy lies higher still in the present case. Physiologists are generally convinced that the rhythmic processes at the bottom of the cardiac pulsations, are based on chemical actions in the cardiac muscle. Leaving apart the founder of the myogenic theory TH. W. ENGELMANN, we mention some authoritative writers, Fano and Borazzi in Ricuer’s Dictionnaire and Hormann in NaGeL’s Handbuch, who embrace this point of view *). Also experimental results may be adduced in support of this theory. SNyDER*) showed that the frequency of the contractions with respect to temperature follows exactly the law, formulated by van ‘tr Horr and Arruenius for chemical reactions *) and experiments, independently made by J. Gewin, entirely confirmed this. *) Whereas the influence of temperature is considerable, that of pressure is very small. This agrees with the small significance of external pressure for so-called condensed systems, i. e. systems in which no gaseous phases occur. ) ) Syyper, Univ. of California Publications IL. p. 125. 1905. ) E. Conen, Voordrachten. Blz. 236 1901. ) J. Gewin, Onderzoekingen Physio]. Lab. Utrecht (5). Dl. VII, p. 222. 5 > > (592 ) For the automatism it seems to me to be settled, that it must be based on chemical processes. For the remaining cardiac properties: conductivity, local sensitiveness to stimuli, contractile power and tonicity the decision is more difficult. The law of van ’r Horr-ARRHRNIUS concerning the relation between reaction-velocity and temperature can only be applied if the duration of the reaction is known. Now the velocity of conduc- tion, measured with this purpose, increases with temperature up to a certain optimum *) whereas correspondingly the duration of the contractions is diminished’). The local excitability, however, has not been studied yet from this point of view, while also for the contractile power the time factor is still lacking. But the contraction of a muscle and also that of the cardiac muscle is so universally considered a truly chemical process, that the reader will not object to classing it among chemical phenomena without further arguments. As to the tonicity we are absolutely in the dark, although we know that rise of temperature chiefly brings about a change, in which the tonus is definitely abolished. In preparing his thesis Mr. Wourrrson had chiefly to deal with : 1. changes in the automatism (chronotropy) ; 2. changes in the contractile power (inotropy). Both these changes are purely chemical phenomena, as was shown above. For chemical processes the law of GuLpBERG and Waar holds *), and we may apply this law to the processes here dealt with. For this purpose we shall have to give a nearer definition for our special case of the conception “times of equal change”. By “times of equal change” we mean the times in which a defi- nite reaction has taken place between two accurately fixed and in the corresponding cases analogous terminal points. The total duration of a cardiac period is such a characteristic time element, the begin- ning and end of which cannot be determined with the balance after chemical analysis, but still are determined by biological characteris- tices. The time between the beginning and the end of a cardiac period may be looked upon as a time of equal change provided no 1) Ta. W. Eneetmann. Onderz. Physiol. Lab. Utrecht (3d series) III p. 98. Above the optimum the conductive velocity diminishes again. 2) HorMANN l.c. p. 247. Recently confirmed by V. E. Nierstrasz; vide acad. thesis, Utrecht 1907, p. 145, fig. 22: a fall in temperature of 9° gave an increase of the duration of the systole to the double value. 3) E. Conen. Ned. Tijdschr. v. Geneesk. 1901, Vol. I, p. 58. Cf. also ZwAARDEMAKER, ibidem, 1906. Vol. II. p. 368. (593 ) inotropic changes oceur*) and the mechanical resistance which the heart has to overcome, has remained the same. These premisses made, we may at once apply the fundamental equation of GurDpBERG and WaAaGE’s law ; Pe kCn Here £ is a constant, the constant of the reactional velocity, C is the quantity of the substance, taking part in the reaction, 2 is the exponent, determining the so-called order of the reaction, while indicates the reactional velocity. About the exponent 7 nothing can be stated a priori for the heart. Toxicological experiments, in which the quantity of the reacting substance diminishes, might perhaps teach us something in this respect; perhaps also experiments on fatigue might give us some clue; at present, however, no data at all are available. Whether there are intermediate reactions and in what number, cannot be ascertained. Under these circumstances | assume, quite arbitrarily, that the present case is the simplest and that the exponent is unity. If later this assumption turns out to be wrong, our calculations will still apply, mutatis mutandis, without losing their meaning. In this simple case the formula runs: gh C. When the vagus is stimulated a very marked alteration of the times of equal change is noticed. The reactional velocity of the hypothetical chemical process, which lies at the bottom of the auto- matism, must consequently undergo a very considerable change. Such a change cannot take place unless either / or C are modified. In the literature on the subject both views are put forth, but only the conception that # changes, leads to a clear explanation without further auxiliary hypotheses. It also fits in best with a recent paper of Martin’), according to which vagus-inhibition is ascribed to the action of K-ions and is counter-acted by rise of temperature. The significance of the ions of the alkalies and alkaline earths for the cardiac muscle is indeed by no means fully explained, even after the mumerous investigations of J. LorB and his followers and critics — they are regarded by some as the cause of the continually excited condition of the cardiac muscle, as the stimulus for the automatism *), by others as the condition, necessary for keeping the 1) In the ventricle vagus-stimulation produces no inotropy. 2) Martin. Amer. Journal of Physiol. Vol. XI, p. 370, 1904 (Martin himself seems to assume a compound of K-ions with C). 3) Wenckepacu. Die Arythmie etc. Kine physiol.-klinische Studie. Leipzig. 1904. 40 Proceedings Royal Acad. Amsterdam. Vol. IX. ( 594 ) active substances in solution ') — they certainly do not enter into simple chemical combination with the cardiac substance, by which this latter would become unfit. If this latter were the case, the life- prolonging influence of Rineer’s solution and the remarkable anta- gonism of Na and K on one hand and Ca on the other, would be entirely unexplainable. By placing the principal weight on the hypothesis that the vagus alters the constant of velocity, of reaction we were led to the applica- tion of the formula for the catalytic acceleration of a chemical reac- tion. The catalytic acceleration is here negative. The explanation of the formula will be found in G. Brepie’s work. It runs: i 1 (, rr t,) (4, ik LN, By application to our experiments, the normal duration of the period being indicated by (t,—t,) the altered one during the principal retardation by (¢, —?,), a relation became evident which appears to be constantly found between the intensity of the vagus stimulus on one hand and the retardation, indicated by 8 on the other. (An examination of the curves, recorded by the heart would show that the retardation affects principally the diastole part of the process, but since for this part, taken separately, the times of equal change cannot be sharply deter- mined, our calculations enclose the whole process). When the vagus stimulus increases the retardation increases also very gradually, until a definite degree is reached; from this moment the reactional velocity of the hypothetical process of the automatism remains the same, independent of any rise in the intensity of the stimulus. Only by increase of the duration of the vagus stimulus, a new retardation may be produced, which is pretty well proportional to the extension of the duration of the stimulus. For a warmed heart all this holds without any alteration. SR 1) H. J. HauBureer. Osmotischer Druck und fonenlehre.” Bd. III, p. 127. (595 ) Exp. 8, VI. 1906. Emys orbicularis. Right nervus vagus stimulated on non-polarising electrodes with charging currents. Capac. of the condensator 1 microfarad. Number of stimuli 10 (2 per second). Between the series of stimuli pauses of 4 minutes; external temperature 18° C. Micro- | Initial Total Ergs | B coulomb retard. in °/, | retard. in °/, | | 0.80 | 3.20 13 23 —0.0392 0.82 3.36 92 143 —0.1662 0.84 3.58 95 133 —0.1694 0.86 3.69 282 347 —0. 2555 0.88 3.87 320 385 —0.2716 0.90 4.05 320 364 —0.2635 0.92 4.23 322 360 —0. 2648 0.94 4.42 346 364 —().2765 0.96 4.61 337 366 —0.2575 0.98 4.80 337 398 —0. 2667 1.00 5.00 343 398 —0.2679 1.04 5.41 333 394 —0.2570 1.08 5.83 346 4A0 —0.2765 4.12 6.27 333 367 —0.2661 1.20 7.20 330 322 —0.2480 1.28 8.19 346 373 —0 .2592 1.36 ER) 336 370 —0.2575 1.52 11.50 343 374 —0.2679 1.68 14.41 360 421 —0.2790 1.84 16.93 340 377 —0.2673 3.68 67.74 971 105 —0.2723 5.52 152.35 371 4A8 —0 2723 7.36 270.85 371 M4 —0 2723 9.20 | 423.20 357 377 —0.2702 11.04 609 . 40 333 347 —0 2661 0.80 3.20 330 343 —0.2654 (596 ) Exp. 15, VI. 1906. Emys orbicularis, Right nervus vagus. Non- polarising electrodes. Charging currents. Capac. 0.2 microfarad. Number of stimuli 10; (2 per second). Resting pauses between the series of stimuli 2 minutes. Experimental animal in 0.6°/, NaCl solution, heated to 28° C. Micro- landden Total | 5 5 cou Ergs able in the second | DE ‚retar- | lombs en ate period | dation | So Oe ple | | | | 0.48 DD 15 | 139 | 294 | —0.0785 | —0.3889 0.496 6.15 20 99 Lao — 0.44 MA SOD 0.504 | 6.35 20 | 439 | 205 | —0.1144 | —0.3889 0.52 6.76 26 152 994 | —0.1404 | —0.4035 0.552 neo | 26 152 | 224 | —0.1404 | —0. 4035 0.616 | 9.48 26 152 | 218 | —9.1404 | —0. 4035 0.744 | 13.83 28 199 270 | —0.1587 | —0.4762 44416) | 81443 21 | A57: | 1270 | —0.1261 | —0.4579 De08 be, Gp 28 157 284 | —0.1587 | —0.4579 Two particulars deserve notice: 1. that the greatest retardation falls not in the second but in the third period. 2. With stimulation with 7,61, 9,48, 13,83 ergs turbulent motions occur in the ventricle, followed by the post-undulatory pause, namely in the first systole after the preliminary retardation. The relation brought to light in both these cases might be explained by assuming with Lanerey that the vagus fibres do not reach the heart directly, but first pass a station of the intra-cardial ganglia. If this be the case the stimulated condition of the prae-ganglionie fibres will only be communicated to the post-ganglionie by contact in the ganglion cells. But then the quantitative coercion of WEBER’s law holds for these ganglion cells and a relation as sketched above is not astonishing. To this conception may be objected that probably with stimulation of the post-ganglionic fibres (in the so-called n. coronarius *) the same relation will be found in its principal features. If on this point not only preliminary, but decisive experiments, will have been made, it will be found that the just-mentioned explanation 1) On the n. coronarius as a post-ganglionic nerve vide J. Gewin, |. c. 82. ( 597%) is untenable. Mr. WorrersoN accordingly gives an alternating expla- nation which, in my opinion possesses some probability, and which agrees with Martin’s hypothesis on the nature of the vagus action. Let us suppose that by the action of the vagus some catalytic substance — say Martin’s A-ions — is produced in the receptive substance of the cardiac muscle, then the above stated quantitative relation will be explained, if we may assume that the substance, produced by vagus action, is only to a limited extent soluble in the medium. For with a small production of the catalyser this latter will be dissolved and will increase the retardation, but when the medium has become saturated with the catalyser, further secretion is without effect. It must further be assumed that the newly formed catalyser is at once removed from the substance by diffusion or is deposited in the form of indifferent compound, for the vagus action is known to cease after a short time. Only when the duration of the stimulus is increased and catalytic substance is again and again produced, the disappearance of the catalyser may be compensated and the retardation may be lasting. The second chemical process we meet in Mr. WorrersonN’s thesis, that of the contractility, cannot be submitted to the above followed treatment, since the time-factor is wanting. We tried to introduce this latter by seeking the relation between the intensity of the vagus stimulation and the duration of the inotropic action, but this latter is not itself a chemical reaction, but only a modification of the conditions under which periodically recurring reactions take place. The negative inotropy may at the utmost be regarded as a diminution of the quantity C in the formula g=/C, which amounts to the assumption that by vagus stimulation the quantity of the just men- tioned substance, undergoing chemical change, is diminished. But this also is uncertain, for in the chemical reaction of the automatism (’ represents part of LaNarrY’s receptive substance, which is different from the contractile substance. So I prefer to keep the two chemism apart and to consider the inotropy entirely by itself. Placing ourselves on this point of view, we notice: 1. that with feeble and increasing vagus stimuli the inotropic effect on the sinus and auricle gradually increases with the intensity of the stimulus, until a certain degree of inotropy has been reached, after which it does not increase further for any intensity of the stimulus; 2. that an analogous relation holds good for the duration of the inotropic effect; 3. that the pessimum of contractility is found about the end of the first third or fourth part of the total duration, for which the inotropy exists. ( 598 ) Summarising we arrive at the following conclusions: A. the chronotropy, produced by stimulation of the vagus, may be reduced to a negatively catalytic action on a chemical process which lies at the bottom of the pulsation. B. the inotropy admits by analogy of a similar interpretation, but it is impossible to prove this, since at present no times of equal change can be determined here. As secondary results we mention: a. the existence of twofold negatively chronotropic fibres in the right vagus of the tortoise. b. a particularly great sensitiveness of the heart of the tortoise for inotropy of the auricle by vagus stimulation, in such a degree that a single condensator discharge may produce the stated modification and that also with cumulative stimulation it appears sooner and lasts longer than the chronotropy. c. the occasional occurrence of spontaneous cardiac turbulence in a warmed tortoise heart, immediately after a principal retardation brought about by vagus stimulation. ER RA TUM: In the Proceedings of the meeting of December 29, 1906. p. 504, line 13 from the bottom: for 2 read 4 p. 511, lime 5 from the top: for 0.052 read 0.104 (February 21, 1907). ae ee eat KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM. PROCEEDINGS OF THE MEETING of Saturday February 23, 1907. eh (Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige Afdeeling van Zaterdag 23 Februari 1907, Dl. XV). Gen EEN DT S: A. P. N. FRANCHIMONT: “Contribution to the knowledge of the action of absolute nitric acid on heterocyclic compounds” p. 600. F. A. H. SCHREINEMAKERS: “On a tetracomponent system with two liquid phases”, p. 607. J. BoËsEKEN: “On catalytic reactions connected with the transformation of yellow phosphorus into the red modification”. (Communicated by Prof. A. F. HorrLeEMaAN), p. 618. J. D. van DER Waats: “Contribution to the theory of binary mixtures”, p. 621. (With one plate). Pu. Kouystamm: “On the shape of the three-phase line solid-liquid-vapour for a binary mixture”. (Communicated by Prof. J. D. vaN DER WAALS), p. 639. Pu. Kounstramm: “On metastable and unstable equilibria solid-fluid”. (Communicated by Prof. J. D. van DER WAALS), p 648. (With one plate). W. H. Krersom: “Contributions to the knowledge of the Z-surface of van per Waars. XIII. On the conditions for the sinking and again rising of a gas phase in the liquid phase for binary mixtures”. (Continued). (Communicated by Prof. H. KAMERLINGH ONNES). p. 660. H. KAMERLINGH Onnes and Miss T. C. Jorres: “Contributions to the knowledge of the L-surface of VAN DER Waars. XIV. Graphical deduction of the results of KuENEN’s experiments on mixtures of ethane and nitrous oxide’, p. 664. (With 4 plates). P. NieuwenNuHurse: “On the origin of pulmonary anthracosis”. (Communicated by Prof. C. H. H. Spronck), p. 673. 41 Proceedings Royal Acad. Amsterdam. Vol. IX. ( 600 ) Chemistry. — “Contribution to the knowledge of the action of absolute nitric acid on heterocyclic compounds.’ By Prof. A. P. N. FRANCHIMONT. (Communicated in the meeting of January 26, 1907). When searching about twenty vears ago for the rules according to which nitrie acid’) acts on hydrogen compounds, not only on those which contain the hydrogen in combination with carbon, but also on those which contain it in combination with nitrogen, I found that the hydrogen combined with nitrogen to the atomic group NH, does not act on nitric acid, when, in cyclic compounds, this group is placed between two groups of CO, but it does act if placed therein between the group CO and a saturated hydrocarbon residue ?), and it may be added: not if placed therein between two saturated hydrocarbon residues, although I have not mentioned this previously. It is a peculiar fact that the hydrogen of the group NH does not act on nitric acid if this group is placed between two similar groups such as CO, or saturated hydrocarbon residues, but it does act if placed between two dissimilar ones; so that it might be thought that a tautomeric form is essential for the reaction. There are, therefore, in reality three rules, which, when considered more closely, apply also to acyclic compounds and which, although the cycle also exerts an influence, appear to spring mainly from the nature of the substance in which the group NH is placed: viz. secondary amine, amide or imide. In acyclic amides it was found that not only the acyl group in particular, but also the alkyl group exerts an influence on the reaction; we may, therefore, expect something similar in the cyclic ones. The first of the above rules was mainly deduced from the behaviour of penta- and hexa-atomic cyclic urea derivatives, but was confirmed also in the case of other compounds. For instance GONE CO—N H | Se | Seo and CO CO CON GONE parabanic acid alloxan 1) Namely the real (absolute) acid which may be obtained by distilling a mixture of nitric acid 1.42 with twice its weight of sulphuric acid at a gentle heat under reduced pressure (Recueil XVI. p. 386). 2) Which, however, weed not be the group CH, as stated wrongly by Harrigs (Annalen 327. p. 358). The pages of the Recueil referred to by him contain exactly the proof of the contrary. I have also never spoken of “héchst concen- trirter Salpetersäüure’’ as he says, and of which he thinks he must “den Begriff festlegen”, and for which he then recommends something which in many cases cannot give a good result. ( 601 ) could be evaporated with nitric acid on a boiling waterbath without suffering any decomposition, CO—NH CH,—CO | Nw and also | N NH The CH, CO Ek Heef CH,—CO CO—NH succinimide malonureide gives a nitroderivative, but with the nitro-group attached to the carbon; the two NH-groups do not act. The second rule is also based mainly on the behaviour of penta- and hexa-atomic cyclic urea derivatives. For instance CH, —NH = Sco pera CH,—NH ethyleneureine gave a dinitroderivative, which on boiling with water yielded carbon dioxide and ethylenedinitramine. To this I may now add: CH,—NH | x CH, CO | a CH,— NH Trimethyleneureine of which I have stated recently with Dr. FriepMANN that it gives directly a dinitroderivative, which on boiling with water yields carbon dioxide and trimethylenedinitramine. CH,—NH | ‘co and its methyl derivatives CO—NH Hydantoin CH,—CH—NH (CH,), C —NH CH,—NH | = Sco keset 21 SAG Bes Wikis fel ARO es CO—NH CO—NH CO—N.CH, a lactylurea acetonylurea 1 Nmethylhydantoin gave mononitroderivatives, which on boiling with water were decom- posed with evolution of 1 mol. of carbon dioxide and formation of a nitramino-amide; for instance nitrohydantoin yields nitramino- acetamide. 41* ( 602 ) CH,—NH To this I may add the CH, CO recently investigated with Dr. A CO —NH hydro-uracil (8 lactylurea) FRIEDMANN, which yields, equally readily, a mononitroderwatiwe'), which on boiling with water yields, in an analogous manner, carbon dioxide and B nitraminopropionamide, from which we have prepared B nitraminopropionic acid, also its barium and silver salt. This decomposition proves the position of the nitrogroup, and at the same time these substances are all a confirmation of the first rule because the group NH, which is placed between the two CO-groups, has not taken part in the reaction. CH, GES ES beu sl | CH, CO and CH, CO Fl | | CH,—NH CH,.CH— NH a Piperidone a methylpyrrolidone gave with nitric acid N,O, presumably derived from a nitro-compound unstable towards nitric acid at the ordinary temperature ; for it has been shown that some nitramides are decomposed by nitric acid at the ordinary temperature with evolution of nitrous oxide; whilst others may be evaporated with this acid on a boiling waterbath with impunity. The rule was confirmed five years ago with cycles in which oxygen takes part, for instance NH NH—CH ZOE a | ne : CO | and CO CH, aN | ke O——CH? O—CH, u. célo tetrahydro-oxazole. wu. céto pentovazolidine gave on evaporation with nitric acid, mononitroderivatives, which on ') Tare, stated about this substance (Ber. d. D. ch. G. 33 p. 3385) that it is not affected even by prolonged boiling with concentrated nitric acid; evidently he has not used absolute nitric acid. ( 603 ) boiling with water were decomposed with formation of carbon dioxide and a nitramino-alcohol. CH,—CH, The third rule is derived from the behaviour of CH, CH, which | | CH,—NH piperidine yields with nitric acid a nitrate, but not directly a nitro-compound. This, however, may be prepared from a number of piperidides, to which we added recently the piperidides of sw/phuric and succinic acids, or from the nitrate with acetic anhydride as found by BAMBERGER. CH, —NH—CH, I have noticed recently that | | behaves in the same CH,—NH—CH, piperazine. manner. The above cited new investigations and those which follow origi- nated in a research by Mr. A. Dork. He had prepared for practice CH,—NH—CO | and we treated this with nitric acid. But even CO— NH—CH, glycocol anhydride on evaporation on a boiling waterbath it gave no evolution of nitrous oxide, no nitroderivative, but a nitrate. I had expected this CH,—_NH—CH, behaviour sooner from the unknown | | which is one of its | CO— NH—CO iminodiacetic imide isomeres, and in which one NH-group is placed between two CO-groups and the other between two saturated hydrocarbon residues, but not from glycocol anhydride in which each NH-group is placed between CO and a hydrocarbon residue, and about whose structure no doubt could be entertained. At most, we might suspect here a tautomer which does not react with nitric acid, or in all other cases in which nitric acid does act we might assume a tautomer and not here. *) Mr. Donk’s nitrate, a very loose compound, appeared to be a mono- nitrate, and on applying BAMBERGER’S method for amines (treatment 1) Harries l.c. suspects in 1 N methylhydantoin a tautomer CHN, | ‘OH Nets CO —NCH; which, however, yields with nitric acid the same nitromethylhydantoin. ( 604 ) of the nitrate with acetic anhydride) he obtained a mononitroderivative, of which he proved the structure by acting on it with methyl- alcoholic potassium hydroxide, which yielded a properly crystallised acid, namely NO, NH CH, CO NH CH, CO, H. nitraminoacetylaminoacetic acid The reaction therefore took place as in all other cases where NO, and CO are both linked to a nitrogen atom; by absorbing the elements of water H and OH the group CO leaves the nitrogen whilst NO, remains attached to it. After the departure of Mr. Donk, who did not wish to prosecute this matter, Dr. FRIEDMANN took it up and obtained the dinitro-compound from glycocol anhydride by treatment with excess of nitrie acid and acetic anhydride. By the action of ammonia on dinitroglycocol anhydride nitroaminoacetamide was obtained, and by means of sodium hydroxide nitraminoacetic acid was formed in such a quantity that the formation of two molecules was no longer doubtful. The position of the two nitro- groups on the nitrogen atoms has, therefore been sufficiently proved. CH,-CH—NH—CO | | when evaporated with nitric acid also CO—NH—CH-CH, Alanine-anhydride gave a nitrate only, which on treatment with acetic anhydride yielded a dinitroalanine anhydride. These results, which formed a first deviation from the rule previously laid down, incited to further research. For it was shown plainly that besides the placing of the group NH between CO and a saturated hydrocarbon residue, the other part of the molecule may also influence the reaction in such a manner that a direct nitration is prevented, even on warming, although nitro-compounds actually exist. The question, therefore, arose as to the behaviour of those isomers of glycocol anhydride, which possess the same atom-groups, but arranged in another order. There may be eleven cyclic compounds which consist of two groups of NH, two groups of CO and two groups of CH,, of whom however three only are described in the literature, namely : CH,—NH—CO CH,—CH, CO CO—NH —CH, | | and! CO —NH-—CH, NE— CON CO—NH—CH, glycocol anhydride hydro-uracil ethyleneoavamide The last one, however, only in an impure condition, as described by Horrmann in 1872, and which we have not yet succeeded in obtaining in a pure state. This substance had a special importance. It has the two grcups ( 605 ) NH, also between CO and CH,, and, according to the rule, it ought to yield readily a dinitro-derivative; either stable or unstable. Still it might be that it was not attacked at all by absolute nitrie acid, for if we remember that diacetamide, although slowly, still evolves N,O with nitric acid and, therefore, presumably forms an unstable nitro-compound under those circumstances, and if we compare this with the cyclic succinimide, which is not attacked at all even on warming and which is connected with it in such a manner that it contains two hydrogen atoms less, and thus causes the cyclic combi- nation, one feels inclined to attribute to the cyclic combination the prevention of the action of the nitric acid. We might also compare ethyleneoxamide to dimethyloxamide which is readily nitrated, and is related to it in the same manner as diacetamide to succinimide, and if the cycle formation has the same effect here as it has in the other case, ethyleneoxamide should not be attacked. Preliminary experiments with the impure substance showed that no stable dinitro-derivate appzars to be formed; at most, one which is at once decomposed by nitric acid, or it is not attacked at all. A very slow evolution of N,O and CO, takes place, but this may be due to the impurity. Of the eleven possible isomers there are only two urea derivatives namely hydro-uracil, which, as stated, conforms to the rule and gives CH,—CO—CH, a mononitroderivative. The second is | | . RiiGHEMER NH —CO—NH acetoneureine thought in 1892 that he had obtained this substance by the action of chloro-formic ester on diaminoacetone, but it was merely a surmise; no analysis was made and the properties were not investigated ; and from our investigations it is extremely doubtful whether he had this substance in hand, for although we made the experiments in various ways we could obtain nothing else but acetondiurethane, from which a dinitro-derivative was readily obtained. A number of other methods for preparing acetonureine from diaminoacetone were tried but always without good result. In the meanwhile we are continuing our experi- ments for, we attach great importance to this substance as a second urea derivative, seeing that the first one conforms to the rule. CO—CH,—CO A fifth isomer would be | | which we have tried in NH—CH,—NH methylenemalonamide vain to prepare from malonamide and formaldehyde. In this case ‘it is the group CH, of the malonic acid which appears to react ( 606 ) principally; but even with the amide of dimethylmalonie acid and formaldehyde we have not arrived at the desired result. Methylene- malonamide is of importance for this reason, that the CH,-group of malonic acid might give a nitroderivative, whilst this may be equally expected from the two NH-groups. CH,—NH—CH, A sixth isomer is the already quoted | | of which one CO —NH—CO iminodiaceticimide might expect that it should yield with nitric acid only a nitrate, but not a nitro-derivative. On heating the diamide of iminodiacetic acid in vacuo, Mr. JoNGKEES obtained a substance which sublimes and has the composition of the imide. This, however, does not behave as was expected, but when evaporated with nitric acid, seems to give a nitro-derivative, whose properties are, however, somewhat different from the usual ones of nitramines or nitramides. The last isomer of some significance for the problem under con- sideration, for the preparation of which no experiments have, as yet, CO—NH —CH, been made, would be | | , in which one NH-group between CO—CH,—NH CO and CH, renders probable a nitro-compound, whereas the second, placed between two CH,, could only yield a nitrate. The other four are derivatives of hydrazine, and are of no importance for our problem, because the two NH-groups contained therein are CH,—CO—NH in a state of combination. One of those | | has been pre- CH,—CO—NH pared by Dr. FrIEDMANN and, when it was brought in contact with nitric acid a violent evolution of red vapours was noticed, evidently caused by oxidation. The details of these researches which of course, are being continued will appear in the “Recueil des Travaux chimiques des Pays-Bas.” But it is evident that the second rule will have to be altered, namely in that sense that the direct nitration (if any) of the hetero- eyelie compounds, which contain NH placed between CO and C,H,, depends also on the manner in which the groups, between which the group NH is placed, are combined; therefore it is the same as has been noticed with acyclic compounds. In how far the cycle itself plays a rôle has not yet been satisfactorily made out but we may point, provisionally, to one peculiarity, namely, that the three com- pounds which do not seem to conform to the previously established rule contain the NH-groups in the para position in regard to each other. ( 607 ) Chemistry. — “On a tetrucumponent system with two liquid phases.” By Prof. F. A. H. SCHREINKMAKERS. (Communicated in the meeting of January 26, 1907). Although in the systems of three components with two and three liquid phases there may occur many cases which have been predicted by theory, but have not yet been realised by experiment, I have still thought it would be as well to investigate a few systems with four components to have a glance at this as yet quite unknown, region. I will now describe more fully a few of those systems built up from the substances: water, ethyl alcohol, lithium sulphate and ammonium sulphate. We may represent the equilibria with the aid of a regular tetra- hedron as in Fig. 1; the angular points represent the four components : Li W Am Fig. 1. W = water, A = alcohol, Li = lithium sulphate, Am — Ammo- ( 608 ) nium sulphate. The side AW being invisible has been left out, also the side Lz Am. Li,SO,.H,O and the double salt LiNH,SO, may also occur as solid phases besides Li,SO, and (NH,),5O,. The first is represented by a point Zon the side LW, the second by a point D, not indicated, on the side Lz Am. The equilibria occurring at 6°5 are represented schematically by Fig. 1. The solubilities of the (NH,),SO, and of the Li,SO,. H,O in water are indicated by the points a and e; point c indicates the solubility in water of the double salt and must, therefore, be situated on the line WD (the point D is on the side Zi Am). As Li,SO,, (NH,),S0, and LiNH,SO, are practically insoluble in alcohol, their solubility may be represented practically by the point A. The curve aA is the saturation line of the (NH,),SO,; it indicates the aqueous-alcoholic solutions which are saturated with solid (NH,),SO, . The aqueous-alcoholic solutions saturated with Li,SO, and Li,SO.H,O, are represented by the curve ed which, however, must show a discontinuity in the immediate vicinity of the point A, for the curve consists of two branches, of which the one to the right indicates the solutions saturated with Li,SO,.H,O and the one to the left those saturated with anhydrous Li,SO,. The equilibria in the ternary system: water, lithium sulphate and ammonium sulphate are represented by the curves ab, bcd and de, which are situated in the side plane of the tetrahedron. ab is the saturation line of the ammonium sulphate, hed that of the double salt LiNH,SO,, de that of Li,SO,.H,O. In my opinion, however, this latter is not quite correct, for, according to several analyses, Lithium sulphate seems to mix with the ammonium sulphate, although only to the extent of a few per cent, so that branch de indicates solutions saturated with mixed crystals. As, however, I have not accurately investigated this mixing, ' will continue to speak in future of lithium sulphate monohydrate Li,SO, . H,O. Let us now look at the equilibria in the quaternary system. The surface Am or Aabb,h,b,A represents solutions saturated with solid ammonium sulphate; surface D or Ab,k,b,bcdA represents the solutions saturated with LiNH,SO,; the curve Ac of this surface has as pecial significance, because it indicates the solubility of LiNH,SO, in aqueous- alcoholic mixtures. The points of the surface D facing the curve Ac represent solutions which, in relation to the double salt, contain an excess of (NH,),SO,; the points behind this line show solutions containing an excess of Li,SO,. ( 609 ) The curve Ac must therefore, be situated in the plane passing through AW and the point D of the side Li Am. The surface Li or Ade indicates the liquid saturated with Li,SO, or Li,SO,H,O, or with the above mentioned mixed crystals; it must, therefore, consist of different parts which however, are not further indicated “in the figure. At the temperature mentioned here (6°5) systems of two liquid phases may occur also; in the figure these are represented by the surface LL, or 6,K\b,K, which we may call the binodal surface; this binodal surface is divided by the line A,A, into two parts L, and L, in such a manner that each point of £, is conjugated with a point of L,. Two conjugated points indicate two solutions in equilibrium with each other: with each solution of the surface L, a definite solution of the surface £, may be in equilibrium. Instead of a critical point, such as occurs with ternary mixtures at a constant temperature and pressure, a critical line is formed here, represented by A,A,. Each point of this line represents, therefore, a solution which is formed because in the system of two liquid phases Z,-+ L, the two liquid phases become identical. Let us now look at the sections of the different surfaces: Ad then represents the solutions saturated with LiNH,SO, as well as with Li,SO,H,O; Ad, and 6,6 indicate the liquids saturated with LiNH,SO, and (NH,),SO,. The intersection of the binodal surface with the surface Am namely, the curve b,K,b, indicates the system: L, + L, + (NH,),SO, namely, two liquid phases saturated with solid ammonium sulphate. With each point of the curve D,‚K, a point of 6,4, is conjugated. Each liquid of 6,A, may, therefore, be in equilibrium with a definite liquid of 6,4, while both are saturated with solid (NH,),SO,. The intersection of the binodal surface with the surface D, namely, the curve 0,4,6, represents the solutions of the system L, + L, + Li NH,SO,. With each liquid of 6,4, another one of b,k, may, therefore, be in equilibrium while both are saturated with solid Li NH,SO,. The points of intersection 6, and 6, of these two curves give the system: L, + L, + (NH,),5O, + Li NH,SO,, namely two liquids both saturated with ammonium sulphate and lithium ammonium sulphate which may be in equilibrium with each other. The points £, and &, have a special significance; both are critical liquids which, however, are distinguished from the other critical liquids of the critical curve 4,4, in that they are also saturated with a solid substance: 4, is saturated with ammonium sulphate and #, with lithium ammonium sulphate. It the temperature is raised the heterogeneous sphere is extended; ( 610 ) at about -+ 8° the point #, arrives on the side AWAm, so that above this temperature a separation of water-aleohol-ammonium sulphate may occur in the ternary system. I have further closely investigated at 30° the equilibria occurring in this quaternary system; the results are represented by the schematic figure 2. The saturation surface Am which at 6°.5 still consists of a coherent whole, now consists (experimentally) of two parts separated from each other: this is because the binodal surface 1,2, now terminates on the side plane AWAm in the curve a,k,a,. Li Am Fig. 2. Of the critical line 4,4, the terminal point 4, represents a ternary eritical liquid; all other liquids of this line are quaternary critical ones, of which #, is saturated with solid lithium ammonium sulphate. The phenomenon of the existence of a second heterogeneous region at this temperature was quite unexpected; it is represented in the figure by the binodal surface L,'Z,* or d,k,d,4, with the critical line hk. 1 have not further investigated at what temperature this is formed; it is sure to be present at about 18°, ( 611 ) The binodal surface L, L, intersects the saturation surfaces Am and D; we have, therefore, one series of two liquid phases, saturated with solid (NH,),5O,, and one series saturated with solid LiNH,SO,. The binodal surface L,'Z,' intersects the two saturation surfaces D and Zi. We have, therefore, one series of two liquid phases saturated with LiNH,SO, (curve k‚d, and #,d,), and one series saturated with Li,SO,.H,O (curve k‚d, and k,d,). By d, and d, are represented two liquid phases which are in equilibrium with each other and saturated with LiNH,SO, and Li,SO,.H,O. Of the series of the critical liquids represented by the curve kk, £, is saturated with LiNH,SO, and &, with Li,SO,.H,O. The curve Ac which indicates the liquids saturated with LiNH,SO, without any excess of either of the components runs between the two heterogeneous regions. From this it follows that this double salt at 30° cannot give two liquid phases with water-alcohol mixtures. We, therefore, have at 30° the following equilibria in the quaternary system. liquids saturated with i> (NH )).S0, , represented by the surface Am 2. LiNH,SO, , 3 Soe ae ois D 3, 1,50. H.0; Ke eee 4 fel fh 4. (NH,),SO, and LiNH,SO, ,, „ the curves: bb, and 0,A 5. Li,SO,H,O and LiNH,SO, ,, cae be dd, and d,A system of two liquid phases : 6. in itself represented by the surface L,L, aor is ey PES Ow Oe 8. saturated with (NH,),SO,, represented by the curves a,b, and a,b, a: re NE DO i a rt Kate en 10. 5 ae OENBESO St an ende ee am Rn EA. - sa, ROUTE. - =. pee one Oe a two liquid phases saturated with: 12. (NH,),SO, and LiNH,SO,, represented by the points: 6, and 5, 13. Li,SO,H,O and LiNH,SO,, Ee ee, d, and d, critical liquids : 14. one series represented by the curve K,K, KR fp EE tay ret, 16. one critical liq. saturated with LiNH,SO,, represented by the point K, ROAS Lait, 2 a LNH SO, 3 del A nik EE eee een Y, Lis, HO, ( 612 ) On raising the temperature over 30° the two heterogeneous regions gradually approach each other and finally unite; at what temperature this happens has not been determined, but from the experiments it is shown that this is already the case below 40°; I have also not been able to determine whether this point of con- fluence is situated in front or behind the curve Ac, or perhaps accidentally on the same. I have closely investigated the equilibria occurring at 50° and represented the same by figure 3; any further explanation is super- De Fig. 3. fluous. I must, however, say something as to the points cj and c,, namely the intersecting points of the curve Ac with the saturating curve of the two liquid phases: 6,d, and 6,d,. At first sight we might think that these two liquids may be in equilibrium with each other. That possibility, of course, exists. Suppose we take a water- alcohol mixture of such composition that two liquid phases oecur on saturating with LiNH,SO,. Both liquids will now contain Li,SO, ( 613 ) and (NH,),SO, and it is evident that two cases-may occur. It may be that the two liquids contain the two components in the same propor- tion as they occur in the double salt; it is then as if the double salt dissolves in both liquids without decomposition. If this is the case the liquids c, and c, will be in equilibrium with each other. The second possibility is that one of the liquids has in regard to the double salt an excess of Li,SO, and the other, therefore, an excess of (NH,),5O,; in this case, c, and ec, cannot be in equilibrium with each other. The experiment now shows such to be the case. When I saturated a water-aleohol mixture with ENE SO) at" 56", the alcoholic layer contained a small excess of Li,SO, and the aqueous layer a small excess of (NH,),SO,. From this it follows that the conjugation line does not coincide with the surface DA W but interseets it; the part to the right of the line must be situated in front of the plane and the left part behind it. The alcoholic solution c, of the double salt cannot, therefore, be in equilibrium with the aqueous solution c, of this double salt, but may be so with a solution con- taining an excess of (NH,),SO, . Chemistry. “On catalytic reactions connected with the transformation of yellow phosphorus into the red modification.” By Dr. J. BÖRSEKEN. (Communicated by Prof. A. F. HoLLEMAN). (Communicated in the meeting of January 26, 1907). I. From the researches of Hirrorr (Pogg. Ann. 126 pag. 193) Lemos (Ann. Ch. Ph. [4] 24. 129) Troosr and Haureruuire (Ann. Ch. Ph. [5] 2 pag. 153), R. Scumnck (B. Ch. G. 1902 p. 351 and 1903 p. 970) and the treatises of Naumann (B. Ch. G. 187 2p. 646), SCHAUM (Lieb. Ann. 1898. 300 p. 221), Wuescurmper and Kavrier (Cent. Blatt 1901 I p. 1035) and Roozrsoom (Das heterogene Gleich- gewicht I p. 171 and 177) it appears highly probable that red phos- phorus is a polymer of the yellow variety, which polymerism is, however, restricted exclusively to the liquid and the solid conditions : the vapour (below 1000°) always consists of the monomer P,. From the above considerations it moreover follows that the yellow phosphorus is metastable at all temperatures below the melting point of the red phosphorus (630°); it may, therefore, be expected that it will endeavour to pass into the red variety below 630°. ( 614 ) Although there are many instances where a similar transformation, as with phosphorus at a low temperature, proceeds exceedingly slowly, the velocity in this case is certainly strikingly small. Even at 200°, when the metastable substance possesses a considerable vapour tension, it is still immeasurably small even though red phosphorus may be present. *) This extraordinary slowness, notwithstanding the considerable heat quantities liberated during the transformation, and the complete alteration of properties caused thereby, have a long time since esta- blished the conviction that the two modifications of phosphorus are each other’s polymers and that the red one has a much more com- plex molecule than the yellow one, but the real cause of that slowness is not elucidated thereby. As regards the question how this condensation takes place, ScHENCK (l.e.) was the first to endeavour to answer this experimentally. On boiling yellow phosphorus with an excess of PBr,, he succeeded in changing it to the red modification at 172° with measurable velocity; and from his first investigations he concluded that the order of this reaction was a bimolecular one: AE. This was meant to represent the first phase, for ScHENcK pointed out that red phosphorus had no doubt a higher molecular weight than P,, which subsequent condensation should then take place with great velocity; in other words he arrived at the rather improbable result that the condensation of P, to P, would take place much more rapidly than that of the simple P, molecules to P,. At a repetition of these measurements with one of his pupils (EK. Buck), they came indeed to the conclusion that the reaction is monomolecular (B. Ch. G. 1903 p. 5208). He remarks “Daraus geht mit Sicherheit hervor, dass die Reaction der Umwandlung des weissen Phosphors in rothen monomolekular verlauft.” He, however, adds “Daraus könnte man den Schluss ziehen, dass die Molekular-gewichte des weissen und rothen Phosphors identisch sind.” It strikes me that ScneNcK arrives here at a less happy conclusion. From the occurrence of a mono-molecular reaction we need not necessarily come to the conclusion that the entire process proceeds in this manner. 1) RoozeBoom (lc) compares this to the retardation of the crystallisation of strongly undercooled fusions as 200° is more than 400° below the melting point of red phosphorus: I am, however, of opinion that this view is untenable on account of the relatively high temperature, and particularly the very great mobility of the yellow phosphorus (RoozeBoom l.c. p. 89). The cause of the phenomena must be looked for elsewhere. ( 615 ) On the contrary as in so many other chemical transformations, we must assume that the measurements executed only apply to a subdivision of the reaction, namely to that with the smallest velocity. In this case it is only natural to suppose that the velocity deter- minations of Scmrrek and Buck apply to the decomposition of the P, molecule’) into more simple fragments (P, of P), then at once condense to the red modification so that we may represent the whole process in this manner for instance: P iolyellow ys. 2b. hw ete Ol, nP ae Pe oe tt ta) ten 2 in which the reaction velocity of (2) is very much larger than that of (1). (We might also suppose, as a primary reaction the transformation of the metastable phosphorus into a labile P,; this, however, I do not think so probable because, in the determination of the vapour density above 1000°, a splitting has been indeed observed). It cannot be a matter of surprise that this decomposition velocity at 200°, (without catalyst) will still be extremely small, looking at the great stability of P, in the state of vapour; and if this decom- position, as I suppose, must precede the condensation, the separation of the red phophorus at that temperature will proceed at least equally slowly. There is also nothing very improbable in the very rapid transfor- mation of the dissociated P, or P into red phosphorus. The fact that the allotropic transformation takes place particularly under the influence of sunlight is certainly not in conflict with the idea of a primary splitting, as we know that the actinic rays accelerate the decompositions (such as of HJ, AgBr, C,J,, etc). I wish also to point out that a primary splitting is also accepted in other monomolecular reactions, such as in the decomposition of AsH, (van ’r Horr’s Vorlesungen), of CO (Scuunck B. Ch. G. 1903 p. 1231 and Smits and Worrr. (These Proc. 1902 p. 417). ®) The monomolecular splitting of CJ, into C and CJ, SCHENK and 1) Although the size of the molecule of the liquid yellow phosphorus is not known with certainty, the identity with that of the vapour is however very probable; for the rest it does not affect the argument. *) I omit purposely the beautiful researches of M. Bopensrein, although for the union of S and H. he also arrives at the conclusion that a primary splitting of the Sg molecule precedes the union with Hg, because we are dealing here with heterogeneous systems in which solubility velocities play an important rôle. It is not impossible, that in all cases in which amorphous substances separate we are dealing with such solubility velocities. 42 Proceedings Royal Acad. Amsterdam. Vol. IX. ( 616 ) SirzeNporrr B. 1905, p. 3459, may be interpreted in the simplest manner by the succession of the reactions : OF SO OT, 0 aa eee nC => Cn and '2'Cl =) fae eee i The measurements of ScHENcK and Buck have been made at the boiling point of PBr,. As this is situated at 172°, it appears that the solvent exerts a considerable accelerating influence on the transfor- mation, as pure yellow phosphorus at 200° remains practically unaltered. The solvent, therefore, acts catalytically ; a still more powerful influence has AICI,. If this is brought together with phosphorus in vacuum tubes, the transformation takes place even below 100°, The catalyst is at once covered -with a layer of pale red phos- phorus, which it is rather difficult to remove by shaking, so that it is necessary to add now and then a fresh quantity of AICI,. The action proceeds much more regularly if benzene (and particularly PCI) is added as a solvent. At the boiling point of this, the trans- formation is completed after a few hours (respectively, minutes) ; the product is ScHENck’s scarlet-red phosphorus but much contaminated with benzene and condensation products, which are retained with great obstinacy. In connection with the explanation in part I. I believe that the observations of ScHenck and of myself throw some light on catalytic actions in general. For it is very probable that in this allotropic transformation a splitting occurs first; we notice that the transformation, consequently the splitting, is accelerated by PBr, or AICI,. Will this not occur generally in catalysis? As a dissociation precedes most reactions it is probable that this question must be answered in the affirmative. (I wish, however, to lay stress on the fact, that in answering this question we do not penetrate into the real nature of catalysis. The reason why the dissociation acceleration occurs, whether this is connected with a temporary combination of the catalyst with the active molecules, or whether the catalyst removes the cause which impedes the dissociation, remains unexplained and need not be discussed here any further.) As far as I have been able to ascertain, this conception is not antagonistic to the facts observed; in fact a number of cases are known where a catalyst causes directly a splitting or considerably accelerates the same. ( 617 ) Platinum, for instance, powerfully accelerates the decomposition of hydrogen peroxide, ozone, nitric acid, hydrazine ete. Aluminium chloride causes a direct splitting of the homologues of benzene, of the very stable polyhalogen derivatives, of aromatic ethers, of sulphuryl chloride, ete. The number of these decomposi- tions is so considerable that, in other cases where we cannot prove a direct dissociation by the catalyst, we may still argue that it takes place primarily, or rather that an already present but exceedingly small dissociation is accelerated in such a manner that a system attains the stable condition of equilibrium much sooner than without the catalyst. The great evolution of heat in the process HCCI, + 3 C,H, + (AICI,) = (C,H;), CH + 3 HCl + (AICI,) points to the fact that the system to the right is more stable than that to the left. I attribute its slow progress when no AICI, is used to the small dissociation velocity of chloroform : the catalyst accelerates this dissociation so that the stable condition of equilibrium is attained in a short time. This reaction gets continuously more violent (the temperature being kept constant). This phenomenon may be readily explained if we bear in mind that the reaction proceeds in different stages (C,H, CHCl,, CHCI(C,H,;), and CH(C,H;), are formed in suc- cession) and that the chlorinated intermediate products are decomposed | much more readily than CHCl,. If sulphur is boiled with benzene and aluminium chloride we obtain almost exclusively (C,H,), 5, (C,H,), 5, and H,S. Without the catalyst hardly any action takes place because the dissociation of S, in benzene solution at 80° is negliglible: (if sulphur is boiled with toluene H,S and condensation products are formed without AICI, being present) the alumi- nium chloride accelerates the reaction 5,—>45,, and consequently the formation of the condensation products. This explanation is therefore quite the same as that given for the reaction of P, with benzene and aluminium chloride; the sole difference is that in the latter the second stage of the reaction consists exclusively in the condensation of P, to red phosphorus, a condensation to which sulphur does not seem. to be liable to the same extent, so that the dissociated sulphur forms with benzene the above products. I consider the formation of a compound of the catalyst with one of the reacting substances of importance for the taking place of the reaction in so far only that one phase can be formed ; otherwise it rather obstructs the reaction, because the catalyst becomes to a certain extent paralysed. One of the most powerful catalysts, platinum, is actually characterised because it does not (or at least with great 42* ( 618 ) difficulty) unite with the reacting molecules, but forms a kind of solid solution. Carbon tetrachloride which forms no compound with aluminium chloride is certainly attacked by benzene in presence of that catalyst not less easily than benzoyl chloride which does form an additive product; whilst also the chlorine atom in the acid chloride is certainly not less “mobile” than that of CCL, GusravsoN imagines that the formation of compounds, such as C,H, (C,H,), Al,Cl, is necessary for the action of C,H,Cl on benzene ; these were separated from the bottom liquid layer which forms during the action of C,H,Cl on benzene and aluminium chloride ; if, however, the formation of this layer is prevented as much as possible, the yield of ethylated benzene improves. Therefore I do not call its formation necessary. That it may act favourably perhaps is because the catalyst and also the two reacting molecules are soluble in the same, thus allowing them to react on each other in concentrated solutions. As has been observed above, there is something unsatisfactory in assuming intermediate reactions in order to explain catalytic phenomena. I will try to explain this matter more clearly. As is known, we may express the reaction velocity of a condition change by the ratio: pate a in which the impelling i : resistance force for that change in condition possesses a definite value which a catalyst cannot alter in the least; the resistance, however, is dependent on influences for the greater part unknown. Therefore, the resistance must be lessened by the catalyst and the question to be solved is: “On what does this decrease in resistance depend?” If we suppose that intermediate reactions take place we divide the process into a series of others of which each one considered by itself is propelled by a force less impelling than the total change; the resistance of each of those division processes must, therefore, be much less, and the question then becomes: How is it that those intermediate reactions proceed much more rapidly than the main reaction? which is in fact nothing else but a cireumlocution of the first question: how is it that the catalyst decreases the original resistance? Therefore, by assuming intermediate products, we have not been much enlightened, on the contrary we have made the problem more intricate, because, instead of having to account for a single increase of velocity, we have to look for that of at least two. I call to mind the theory of OsrwaLp who supposes each process io be a succession of condition changes, which will be all possible if they occur with potential diminution. If, however, the first of those ( 619 ) changes can commence only with absorption of free energy, the process will not take place unless a catalyst is added; this, therefore, opens another road . . . Now, in my opinion too much attention is paid to the milestones on that road and too little to the opening itself. This is chiefly caused by the fact that we know so little of the so-called „passive resistances’’, for instance we cannot give a satisfactory explanation of the fact that iodine acts much more rapidly at low tempe- ratures on metals than does oxygen, although the potential decline is much smaller. Still, 1 think that’ we must look for this mainly in the ready dissociation of the iodine molecuie, always supposing that atoms react more rapidly than molecules, a supposition, moreover nearly a century old. If this should be so, the action of a catalyst must be sought for in the increase of this dissociation. Now, we know of a number of reactions where the catalyst forms undoubtedly a compound with one of the reacting molecules, which additive product then reacts with the second molecule to form the final product, with liberation of the catalyst, but even in such a case, which is called by many “pseudo-catalysis” (Wacner, Z. Phys. Ch. 28 p. 48), I do not consider the formation of this compound as some- thing essential without which the acceleration would not take place. I certainly do not consider the formation of such an additive product as being without any significance, as it is an indication that the catalyst can exercise a particular influence on one of the molecules ; the real increase of velocity is, in my opinion, due more to that influence than to the formation of the additive product, and in view of what precedes this, that influence consists presumably of an increase of the dissociation (and through this of the active mass). It is, of course, obvious that a catalyst will act all the more energetically when the additive products are ‚more labile. I have already mentioned platinum and now point also to the H-ions with which, the formation of additive products, for instance when accelerating saponification, is far from probable. As a very lucid „example, I mention the different catalytic influence which iodine and AICI, exert on the transformation of yellow into red phosphorus. From the researches of Brodie (Ann. de Ch. Ph. 1853 p. 592) which I have found fully confirmed, a small quantity of iodine can convert a large quantity of yellow phosphorus very rapidly into red phosphorus at 140°. (As in many other cases, there is a limit because the catalyst is precipitated by the colloidal phosphorus formed. The velocity at the ordinary temperature is very small but becomes plainly perceptible at 80°. We are undoubtedly dealing here with a ( 620 ) ease where the catalyst combines with the phosphorus to PL, ; this substance commences at 80° to dissociate measurably [so that its vapour density can only be determined at a low temperature (Troost CR 95 293)| with separation of red phosphorus. We may, therefore give here a fairly positive answer to the question: How is it that the second division process proceeds more rapidly than the original ? Because P,I, dissociates much more rapidly than P,,. But this is after all but a lucky circumstance, the real cause must be sought in the fact that in order to obtain P,I, the P, molecule must be dissociated to begin with. With AICI, I have not been able to find an additive product, only some indications that, besides the allotropic transformation, a trace of PCI, is formed (even with per- fectly dry substances the manometer, after a few hours’ heating to 100°, showed a slight increase of the vapour pressure). The fact that the red phosphorus formed has in a high degree ihe property of coprecipitating the catalyst might perhaps indicate the possibility of a compound being formed between yellow phosphorus and AICI,; from the above it follows that there is a possibility of a certain reciprocal influence’) but I attribute this coprecipitation to the colloid properties of the red phosphorus, which, when obtained from solvents and also under the influence of rays of light, carries with it a certain quantity. But even if an additive product is found, the existence of this substance is no more the cause of the acceleration than it is in the case of PL, On the contrary, I consider the formation of a compound of the catalyst to be a case of “poisoning”, caused by one of the reacting molecules, just as arsenic and prussic acid are poisons for platinum, because in combining with it, they prevent the entrance of O, and H, (respectively SO,); just as ether is a poison for AICI,, because it unites with it to a firm compound, which does not decompose until over 100’, the temperature at which the catalyst again recovers itself. Now, I cannot deny that we have not advanced much further with this dissociation theory (which is also not absolutely novel) for the question is now: How is it that a catalyst accelerates the dissociation? But my object was to point out that the formation (and eventually the admitting of the formation) of intermediate pro- ducts can certainly never lead to an explanation of the catalytic phenomena. 2°¢ Chem. Lab. University, Groningen. 1) I have also found a similar reciprocal influence in the action of C,H; Br on AlClz in which C,H;Cl and AlBr3 are formed; it undoubtedly points to a disso- ciation. ( 621 ) Physics. — “Contribution to the theory of binary mixtures.” By Prof. J. D. vaN perk Waais. The theory of binary mixtures, as developed in the “Théorie moléculaire’, has given rise to numerous experimental and theoretical investigations, which have undoubtedly greatly contributed to obtain a clearer insight into the phenomena which present themselves for the mixtures. Still, many questions have remained unanswered, and among them very important ones. Among these still unanswered questions I count that bearing on a classification of the different groups of y-surfaces. For some binary systems the plait of the y-surface has a simple shape. For others it is complex, or there exists a second plait. And nobody has as yet succeeded in pointing out the cause for those different forms, not even in bringing them in connection with other properties of the special groups of mixtures. It is true that in theory the equation of the spinodal curve which bounds the plait, has been given, and when this is known with perfect accuracy, it must be possible to analysis to make the classification. But the equation appears to be very complicated, and it is, especially for small volumes, only correct by approximation, on account of our imperfect knowledge of the equation of state. Led by this consi- deration I have tried to find a method of treatment of the theory which is easier to follow than the analytical one, and which, as the result proved, enables us to point out a cause for the different shape of the plaits, and which in general throws new light upon other already more or less known phenomena. Theory- teaches that for coexisting phases at given temperature Be dy de d dw three quantities viz. —| — | , ze and p—v Bs | — de Jom Gt) 7 dv Jr da JT must be equal. The first of these quantities is the pressure, which we represent by p; the second is the difference of the molecular potentials or M, u,—M, u‚‚ which we shall by analogy represent by g. The third of these quantities is the molecular potential of the first component, which we shall represent by M, u,. Now the points for equal value of p lie on a curve which is continuously trans- formed with change of the value of p, so that, if we think all the p-curves to be drawn, the whole v,a-diagram is taken up by them. In the same way the points for given value of g lie on a curve which continuously changes its shape with change of the value of q; and again when all the g-lines have been drawn, the whole v,w- diagram is taken up. Both the p-lines and the g-lines have the property, that through a given point only one p-line, or only one ( 622 ) g-line can be drawn. One single p-line, however, intersects an infinite number of iines of the g-system, and every g-line an infinite number of lines of the p-system. One and the same p-line intersects a given g-line even in several points. However, it will, of course, be neces- sary, that if two points indicate coexisting phases, both the p-line and the gq-line which passes through the first point, passes also through the second point. If we choose a p-line for two coexisting phases, not every arbitrarily chosen value for a q-line will satisfy the condition of coexistence in its intersections with the p-line, because a third condition must be satisfied, viz. that M, uw, must have the same value. The result comes to this: when all the p-lines and all the g-lines have been drawn and provided with their indices there is one more rule required to determine the points which belong together as indicating coexisting points. So in the following pages I shall have to show, when this method for the determination of coexisting phases is followed: 1. What the shape of the p-lines is, and how this shape depends on the choice of the components. 2. What the shape of the g-lines is, and how this shape depends on the choice of the components. 3. What rule exists to find the pair or pairs of points representing coexisting phases from the infinite number of pairs of points which have the same value of g, when p has been given — or when on the other hand the value of ¢ is chosen beforehand, to find the value of p required for coexistence. But for the determination of the shape of the spinodal curve the application of the rule in question is not necessary. For this the drawing of the p- and the g-lines suffices. There is viz. a point of the spinodal curve wherever a p-line touches a q-line. We have viz. Py (dv dw d'w (dv dw dv fr ——| — = 0, and fre — —= 0 for pins dv? B + dvda eee dedv \ dz /q a dix” AE de) Py Py dadv dv da? the value — - and for | — the value — ——, and so we may dw de Jg dw dv? dadv write the equation of the spinodal curve: (=) (5) de }p du) So if we are able to derive from the properties of the components of a mixture what the course of the p- and of the g-lines is, we can derive much, if not everything, about the shape of the spinodal curve. And even when the course of these lines can only be predicted qualitatively, and the quantitatively accurate knowledge is wanting, ( 623 ) the qnantitatively accurate shape of the spinodal curve will, indeed, not be known, but yet in large traits the reasons may be stated, why in many cases the shape of the plait is so simple as we are used to consider as the normal course, whereas in other eases the plait is more complex, and there are even cases that there is a second plait. Particularly with regard to the p-lines, it is possible to forecast the course of these lines from the properties of the components. With regard to the g-lines this is not possible to the same extent, but if there is some uncertainty about them, we shall generally have to choose between but few possibilities. THE COURSE OF THE p-LINES. In fact the most essential features of the course of the p-lines were already published by me in ‘Ternary Systems’ — and only little need be added to enable us to determine this course in any dw given case of two arbitrarily chosen components. As p—=—— dp dv JT dp these p-lines to know the course of the curves en == 0) and * rT av d ( D ) =. dx eT The former curve has a continuous liquid branch, and a continuous gas branch, at least when 7’ lies below every possible 7%, when we denote by 7, the critical temperature for every mixture taken as homogeneous that occurs in the diagram. If there should be a minimum value of 7; for certain value of rz, and 7’ is higher than this mini- , it is required for indicating the course of dp mum 7, the curve (3 = 0 has split up into two separate curves. av) xT In either of them the gas and the liquid branch have joined at a value of v=v,. In this case a tangent may then be traced // to dp the v-axis to each of these two parts of the curve (=) == (| Ul rT dp . The second curve | — | =O is one which has two asymptotes, de) .r and which may be roughly compared to one half of a hyperbola. The shape of this curve derived from the equation of state follows from the equation: ( 624 ) db . ; LEN = ay (v—b)? v" If we now always take as second component that with the greatest db . pclae’ ge ; value of 6, so that Re always positive, it appears from the given av d | equation that the curve (2 =— 0 cannot possess points for these UJ yT da . : values of z, for which zn negative. Only at that value of «# for U | which - =0, this possibility begins, but then only if 7 = 0. If av da at ' : ee T has a definite value 3 must be positive, for points of this curve av to be possible. For v=o, = must be — MRTS. And the value of « which satisfies this equation, indicates one asymptote of the discussed curve by a line // to the v-axis. If this asymptote has been drawn, we may think the mixtures with decreasing critical temperature to be placed on its left side. And on the right the mixtures with increasing critical temperature do not yet immediately follow. For a separation between the mixtures with decreasing and those with increasing 77, ee must be ed a only when MRT = = da b da b 7;, would immediately ascend again on the right of this asymptote; but then 7 would have to be chosen so high, that it was 7’/, 7%, and for the present at least we shall choose 7’ far below that limit. That the line «=c, where c has the value which follows from da J db s 3 - ° == MRI 7,’ is an asymptote, is seen when we think the equation C AH dx da ; dp : ; v" da of the curve | — |= 0 written as follows: a AB de) ‚Tr (v—b)? db Ay da ‚da : v : the value of oe becomes larger from left to right, oo increase wv ph A v : from left to right, or = decrease. For the value of «x, following da BRE BD on epee j heed) from Ge MRT FREE is infinite; for larger values of «, = decreases av Lv ) ( 625 ) v da more and more, and as ¥ can never become = 1, because — av cannot become infinite, the curve v= 6 is the second asymptote. So if « is made to increase more and more, also beyond the values which for a given pair of components are possible in order to examine the circumstances which may occur with all possible systems db for which with positive value of — increasing value of 7%, is always Wa 2: dp found, a minimum volume must occur on the curve (2) == 05.6 av) yT ene b= 6 or this point | — |= P de? Jor Now that we have deseribed in general outlines the two curves which control the course of the p-lines, we shall have to show in what way they do so. From follows that to a p-line a tangent may be drawn // w-axis when it dp . . passes through the curve (e) , and a tangent // v-axis, when it u de )yT dp : ; passes through the curve {—]. But though these are important O/T properties they would be inadequate for a determination of the course of the isobars, if not in general outlines the shape of one of these dp Le lines could be given. The line (2) = 0 viz. intersects. the line Av vT dp ; a : (5) =0 in two points, and it is these two points which are of WU LLT fundamental significance for the course of the p-lines. The point of intersection with the liquid branch is viz. for a definite p-line a double point, the second point of intersection being such an isolated point that it may be considered as a p-curve that has contracted to a single point. The surface p=//(#,v) is namely convex-concave in the neighhourhood of the first mentioned point. Seen from below a section // v-axis is convex, and a section // z-axis is concave. A plane, parallel to the v,«-plane touching the p-surface intersects, therefore, this surface in two real lines, according to which p has ( 626 ) the same value. But for the second point of intersection the two sections are concave seen from below — and there are no real lines of intersection. This second point is a real point of maximum pressure. With all these properties, and also with those mentioned before or 2 a . . still to be mentioned, ae assumed to be positive. *) Hij Now the curve p==eonstant passing through the first point of : dp dp , ; intersection which the curves {|—}]=0O and |— ]=0O have in dv) oT de ),T common, is the isobar whose shape we can give, which shape at the same time is decisive for all those following, either for larger or smaller value of p. In the adjoined figure 1 its course is represented. Coming from the left it retains its direction to the fe dp right also in the point of intersection with the curve | — |= 0, v0 the convex side all the time turned to the z-axis till it is directed straight downward in the point where it meets the vapour dp ; branch of the curve | — ]=0. There it has a tangent // v-axis, and UV / oT from there it has turned its concave side to the v-axis. When it dp dv ; : : meets the curve |—|=0O, | —] is equal to 0 for this as for all Pp B bj de Jr da isobars. Passing again through the curve (2). (=) is again infi- dv ),7\ da), nitely large, and pursuing its course, it passes for the second time through the double point, and further moves to the right, always passing to smaller values of 7, till it has again a tangent // to the : | dp axis of x, when it meets the curve (2) = (0 once more, after which it proceeds to larger value of v. It is clear that in the path it describes from the double point till it passes through this point for the second time, it has passed round the point we have called the second point d !) That the characters of the two points of intersection of the curve (Z)=0 ENT dp with the curve co = 0 are different appears among others from this that when dv Jel these points of intersection coincide as is the case when these curves touch each d'p dp. ( dp other, the quantity —— — — dv* dz” dx dv section depends on this quantity being positive or negative. )=o The character of the points of inter- ( 627 ) of intersection with the curve (5) Foe and where maximum pres- sure is found. In fig. 1 some more isobars have now been drawn besides this one. We obtain the course of the isobars for lower value of p by drawing a curve starting from the left at higher value of v, bearing in mind that two p-lines of different value of p can never intersect, because the p is univalent for given value of « dp and v. Such an isobar cuts the curve (3) = (lon the: left: of the av xT dv isobar with the double point in two points, where (=) = oo, then Ak / ) dp : ; dv passes through the curve {|—]=O in a point where | — | = 0, &/vT adt pT and has then also on the right of the said isobar again two points , : dp a of intersection with the curve (7) = 0, in which points of intersection dv JT j dv again | | =e 2) 5 de), An isobar of somewhat higher value of p has split up into two isolated branches. One of them starts on the right at somewhat smaller value of v; further this branch follows the course of the isobar with the loop, but must not cut it. Arrived in the neighbourhood of the double point it is always obliged to remain at small volumes; there 2 dp ; dv E } it meets the curve (Z)= 0, and it has eS Q. From this point it proceeds to smaller volumes, till a new meeting-point with the same curve causes this branch again to turn to larger volumes. But the second branch of this isobar of higher value of p is entirely inclosed within the loop of the loop-isobar. Such a branch forms a closed curve surrounding the point which we have called the second point ‘ ; dp dp 4 of intersection of the curves ()=0 and (Z)= 0. Such a : P : closed branch passes twice through & = 0, and also twice through at}, 2 dp 4 € dv a & = 0, and has again in the first cases | — | =O, in the second D Jr P dv points of intersection | — | == @ dx}, With ascending value of p the detached portion of the p-line contracts more and more, till it has contracted to a single point. So at still higher value of p only one single branch of the p-line remains. ( 628 ) A similar remark must be made for the curves of lower value of p. The smallest value of p for gas volumes is of course p=0O; but this limit does not exist for the minimum pressure of the mixtures with given value of zr. For this we know that also values of p may occur which are strongly negative. For values of p which are negative, the p-line has again divided into two- disjointed portions, viz. a portion lying on the left in the diagram, which is restricted to volumes somewhat larger and somewhat smaller than that of the liquid branch of the curve (3) = 0, and a similar portion lying zx on the right in the diagram. Also on the locus of the points of inflection of the isobars the given diagram can throw light. So it is evident in the first place, dp that between the two branches of the curve (3) = 0 starting 2 from the double point, both on the left and on the right a connected k k ers dv dp series of points is found where (=) — 0. If the curve (Z)= 0 a J Wee itself should possess a double point, which is the case when 7’ has exactly the value of 7, minimum, this locus of the points of inflec- tion of the p-lines passes through this double point, and when the dp curve (3 — 0 has split up into two separate portions, as is the Ora ease for still higher value of 7, then those points of the two portions dv where — =o belong to this locus. It is also apparent from the Av diagram that two more series of points start from the double point, one on the right and one on the left, as locus of the points of inflection, and that these run to smaller volumes. An isobar with somewhat larger value of p than that of the loop- shaped isobar has a tangent // to the z-axis where it passes through dp k ; : ; the curve () — 0. On the right and on the left of that point it AD) y turns its concave side to the z-axis, whereas at larger distance it must again turn its convex side to it on both sides. So there start dv from the double point four branches on which (=) = 0) dias 42 also easy to see that the branch which moves to the right towards ' op smaller volumes, must pass through that point of the curve & =) v v where the tangent is // z-axis. For an isobar which passes through ( 629 ) d 4 : . the curve (3) = 0 on the left of this point, turns its concave side Ty to the z-axis, but when it passes for the second time through the said curve on the right of the point, its convex side. Hence an isobar where these two intersections have coincided, has its point of inflec- tion in the point itself. If we wish to divide all the v,v-diagram into , d*v ; Eve regions where (=) is either positive or negative, it must be N & P : 5 ; dp borne in mind that also the two branches of line Ty ae 0 them- ae fx selves form the boundaries for these regions, because on that line ( dv — | = 0. dz /, pk dea, i: 5 In all this ze 1 supposed to be positive. For on the contrarv © = dp the course of the line & = 0, to which we could now assign : de), an existence on the right of the asymptote which is given. by db da A se lms MR en ne would be directed to the left of this asymptote, v av be 2 Ba when pas should be negative, so if 2a,, could be >a, + a. For as v ; da v = iz, v da p — ————_,, the value of — decreases only, when — increases. v—b ‚db b 7 da MRT— : dx ; If we put a= A + 2 Be + C2’, and so = —= (BH Cv), it appears f , da . that with C negative « must decrease in order to make 7, increase. a For the points of this line p would then possess a minimum for given 2 d Et 3 8 value of v, and so Te would be positive. From this follows then that vy the two points of intersection of this line with the curve (2) =| v x have interchanged roles. The point of intersection with the smallest volume represents then a real minimum of p, and will have the same significance for the course of the p-lines as the second point 4 : d'a of intersection has, when oT v is positive. And the point of intersection ( 630 ) with the smallest volume has now become double point. I have, however, omitted the drawing of this case 1. because most likely the case does not really oceur, and 2. because the drawing may easily be found by reversing the preceding one. There are e.g. with the solution of salts in water cases which on a cursory examination Ma present some resemblance with the assumption negative, but which yet are brought about by influences perfectly different from da the fact of a negative value for ——. av Such a diagram for the case pe negative, though, would quite fall in with the right side of fig. 1. As in the given figure 7; increases with w on the right side, and there is a maximum value 2 a of 7; on the supposition eae fig. 1 might be still extended to the right Ak till such a maximum 7}, was reached. But then we should also have to suppose that a value of rz could exist or rather a mixture for Ma which at a certain value of 2 the quantity z,n reverses its sign. Every region of fig. 1 of certain width which is taken parallel to the v-axis can now be cut out for a, + a, — 2a,, positive, to denote the course of the isobars. Regions on the left side indicate the course of the isobars for mixtures for which with increasing value of 6 the critical temperature decreases — regions on the right side for mixtures for which with increasing value of 6 the critical temperature increases — the middle region with the complicated course of the isobars when there is a minimum 7. The left region would be compressed to an da exceedingly small one if we wished to exclude the case — negative av or = 0. We do so when putting a,, = Wa,a,. On such a suppo- ax sition a minimum Zr is still possible, but the left region must then have an exceedingly narrow width. There is, however, no reasonable ground for the supposition a,a, = a,,?. There would be, if the quantity a for the different substances depended only on the molecular weights, and so a=em* held for constant value of «. If the attraction, just as with Newron’s attraction, is made to depend on the mass of the molecules, and so if we put a, = €,m,?, and also a, = e‚m,°, it appears that ¢, and e, are not equal. If we now put a,,-— Va,a,, we put a, = mm,Ve,e,. What reasonable ground is there now for the sup- Prof. J. D. VAN DER WAALS. „Contribution to the theory of binary mixtures. aes x x Sai x x x Bie de Proceedings Royal Acad, Amsterdam. Vol, IX, ( 631 ) position that if there is a specific factor e,‚ for the mutual attraction of the molecules of the first kind of which we do not know with what property of these molecules it is in connection, and if there is also a perfectly different factor «, for the mutual attraction of the second substance, we must not represent the specific factor for the attraction of the different molecules inter se by «€,,, but by Wee, It is true that this supposition renders the calculations simpler; I had already drawn attention to this in my Théorie Moléculaire (Cont. I, p. 45). But whether the calculations are somewhat more or some- what less easy does not seem a sufficient ground, after all, to intro- duce a supposition which involves that naturally a great number of possible cases, among others also for the course of the spinodal line, ave excluded. If we put all possibilities for the value of a,,, then a { ‘ & Ag - Ay, = €an ‘also. be’ = 0, viz: for = ———. We need not go so far dx Trans Mes however, to give sufficient width also to the left region. THE COURSE OF THE Q-LINES. d The value of a =q is found from the value of yw: av v SS WRT (2D | aes lr +f (| % For z=0O this expression is negatively infinite, for z= 1 it is positively infinite, so that we have g,=—o and g,=-+ o. But it follows also from the equation of state that for all values / d of « the value of f(Z) dv is also positively infinite for the line v=6. Et Jy v It is true that for such small volumes the equation of state MRT Bs: i= RE not accurate when 6 is not made to depend on UT Vv v, and the quasi association in the liquid state is left out of account, and that the conclusion: J (2) dv is infinitely large for v equal to the Lv v v limiting volume, calls for further consideration before we may accept this as an incontestable truth. But it seems to me that simple con- siderations lead to this conclusion. For the limiting volume p is d infinitely great, and if 6 increases with a, (5) is infinitely large na v 43 Proceedings Royal Acad. Amsterdam. Vol. IX. ( 632 ) i) d aah of higher order, whereas ff (=) dv can again diminish the order of & > v infinity by a unit, because the factor of dv has this higher order of infinity only for an infinitesimal value of dv. But still the thesis /d remains true that { (2) dv is infinitely great for v = 6. v So there is strong asymmetry in the shape of the g-lines. Whereas g=—~ holds for =O and every value of v > b,, q = + @ holds all over the line of the limiting volumes, and for all volumes on the line «2—=1 which are larger than },. We derive immediately from this, that all the g-lines without exception start from the point « =0 and v= 6,. In this point the value of q is indefinite, as also follows from the value of g as it is given by the approximate equation of state, viz.: db dv ; de da q = MRT 1 + MRT an 1— v—b v It also follows from the approximate equation of state that at their starting point all the g-lines touch the line v= 6, of course with the exception of the line g=— oo. For we derive from dw\ _ da), d? dv ad? ee a oe its aa dedv) \duJg da* or aw dv\ _ dx? du) q dy dadv ay For En da? the approximate equation of state yields: db db d MRT — MRT ve dy MRT de ae da? dz? —-x(1—z2) v—b ij (v—b)’ ¥ v We already found the value of = = & Don above. For 6 we find therefore: ( 633 ) AEC @ MRT MRT 4% 7 de de? dy @(l—e) ' v—b da? vB? = dage MRT db dal If we multiply numerator and denominator by (w — 6)? and if we db de dv put v=b, we find for the starting point of the g-lines (5) = at gbi at least if we can prove that — — is equal to zero for «=O and & v=b,. To show this, we put b=b, + Be + ye’, and so v—b = v—b = (v — b,) — «8 — yx’, and then we find for (v— 6) — the value: x v—b, ED B vel. dn: f The term — — is indefinite, but nevertheless the given value av multiplied by v — 6 is really equal to zero. This result, too, is still to be subjected to further consideration, because it has been obtained by the aid of the equation of state, which is only known by approxi- mation. And then I must confess that I cannot give a conclusive proof for this thesis. But I have thought that I could accept it with great certainty, because in all such cases where a whole group of curves starts from one vertex of an angle, e.g. for the lines of distillation of a ternary system, I have found this thesis confirmed that then they all touch one side of the angle. Only in very exceptional cases the thesis is not valid. | Moreover, the theses which I shall give for the further course of the g-lines, are independent of the initial direction of these lines. Only, the g-lines themselves present a more natural course when their initial direction is the indicated one than in the opposite case. dv From the value given above for e follows that they have a o/q dp tangent // v-axis, when 7 — 0, and a tangent // z-axis, when Uy Py da? =0Q. Hence they have a very simple shape in a region where d @? the lines ( *) = 0 and — =0 do not occur. Starting from the Cf}. av point == 0 and v=), they always move to the right and towards 43* ( 634 ) dv\ | oS tf 0 larger volume, and { — } is always positive. Therefore | — | and as de), dee) Hi 2 will presently appear, Bae always positive in such a region. As av U v becomes greater the value of g approaches to MRT es and v for very large value of v the g-lines may be considered as lines parallel to the v-axis, for which the distribution over the region from «0 to &=1 is symmetrical. The lines for which q is negative, extend therefore from 2=0O to «= 3 and for #— ¥ the value of g=0. It will only appear later on that yet in their course probably two points of inflection always occur for small volumes, a fact to which my attention was first drawn by a remark of Dr. KounstamM, who had concluded to the presence of such points of inflection in the g-lines from perfectly different phenomena. | . (AP But as soon as the line (=) = 0 is present (the case that also LJA dp da? makes its appearance in the course of the g-lines. A q-line, viz., which cuts this locus, has a tangent //v-axis in its point of inter- section, and reverses its course in so far that further it does not proceed to higher value of «a, but runs back to smaller value — may be =O will be discussed later on), a new particularity dv Ase so that (5 , which is always positive in the beginning, is hence- q forth negative. From that point where they intersect the line dx 7 this quantity becomes smaller negative. Still for v=», the q-line must again run parallel v-axis. So there must again be a point of inflection in the course of the g-line. In fig. 2 this course of the g-lines has been represented, both in the former case when they do v d dv B = 0 and where (Z) may be considered negatively infinite, q not intersect the curve (2) , and when they do so. In the latter DH hos case they have already proceeded to a higher value of « in their course than that they end in. They end asymptotical to a line ==, and at much smaller volume they also pass through a point a= a. The point at which with smaller volume they have the same value of re as that with which they end, lies on a locus which has a _ (ap shape presenting great resemblance with the line (2) = 0. The value U) y ( 635 ) X-axis. ‘ ' 1 1 { Un gaes> renten cece ae = Ga , A Oe wee oe a ‘ EN ast) gana Penia ‘ - A Bae U ' , / 1 ‘ ie ; / , / | 1 ‘ UB) =(G ' / ‘ 1 | ' 1 S ' 1 1 ' , n ee ' 4 ' ' ( Ei ' re i ' 1 y ' 1 ' H t | ! i ' ' ' 7 I ‘ ' ‘ ' 1 1 1 H { t ' ' 1 , ‘ ‘ ' ' ‘ ' Fig. 2. the points of this locus may be derived in the following way If we write p= MRT {A —e) log A—a) + « log x} +f pdv v for dp\ dp then (3) = =o MRE) tn EK At infinite volume the value of g= MR er as we saw above : x The locus under consideration must therefore be determined by rd +@) dv = 0. Hence on the line «= the final value, a point must v ( 636 ) od d be found such that, proceeding along that same eine, (de = 0. EN) v So from this follows immediately 1. that the points of the said locus dp restrict themselves to those values of x2 in which the curve IE — 0 f bh ee occurs, 2. that the points must be found with smaller volumes than d es those of (z) =0. For such points with smaller volume is viz. af yy d a : @ positive, and for points with greater volume negative — Uy however when the volume may be considered as a gas volume this negative value has an exceedingly small amount. And even without men | drawing up the equation f (E) dv = O, we conclude that the said locus GI, ; d, ; : f has the same wz-asymptote as Ls = 0 itself, and is further to be de), found at smaller volumes. Hence it will also have a point where its tangent runs // z-axis. There is even a whole series of loci to be given of more or less importance for our theory, which have a ‚(dp /dp course analogous to that of | — |=0 and fl ~~ \dv—0. dz}, da), v DUO re p The latter is obtained from & by integration with respect to v; ax} y - all the differential quotients with respect to v of the same function dp Vp Ps put equal to O have an analogous course — thus ==) UU / adv Lv which is a locus of great importance for our theory. That it has the same Ip : ‘ xv asymptote as & = 0 itself, and that all its other points are to at be feund at higher value of v, follows immediately from the follow- ing consideration. For a point of the line 5) = 0 the value of du) 7 dx positive — but for points with larger v negative. For v = o this negative value has, however, again returned to 0. So there must have been a maximum negative value for a certain volume larger than that dp & — 0. For points of the same & and smaller v this value is ( 637 ) 2 d? for which this value =0. These are the points for which 0. Lav . ap . For smaller value of the volume ands’ therefore negative — on the avav other hand positive for larger volumes. The approximate equation of state yields for the loci mentioned and for following loci these equations : db da da da dp OE dv = 0 v—b el if dx 7 db da de de d eee (v= Oe any db da de dx d? Gaeta == at for jd =— 0. (v—b)’ v? dadv And so forth. But let us now return after this digression to the description of the shape of the g-lines. Whenever a q-line passes through the locus “(dp LS RP = dv = 0, the asymptote to which it will draw near at infinite av v volume is known by the value of «x for that point of intersection. For the present it does, indeed, pursue its course towards higher value of «, d but when its meets the locus (2) — 0, it has the highest value of Us ov wv, and a tangent //v-axis. From there it runs back to smaller value Of: #. And this would conclude the discussion of the complications in the shape of the g-lines, if in many cases for values of 7’ at which the solid state has not yet made its appearance, there did not exist another locus, which can strongly modify the shape of the g-lines, and as we shall see later on, so strongly that three-pbase-pressure may be the consequence of it. 2 2 = w dw The quantities and — occur in the equation of the spinodal dv? da? curve in the same way. It may be already derived from this that a? ad? s =) ang W : = 0 will have the same dx? the existence of the loci significance for the determination of the course of the spinodal line. ( 638 ) That as yet our attention has almost exclusively been directed to dw dv? given binary mixture furnishes points for the latter locus for values of 7 below 7; for that mixture, whereas the conditions for the dw —0 is due to the fact that we know with certainty that a existence of a locus — 0 are not known and it might be x° suspected that this remained confined to temperatures so low that the solid state would have set in, and so the complications which would be caused by this, could not be observed. That such a sup- position is not quite unfounded may still be safely concluded from the behaviour of many mixtures, which quite answer to the consi- 2 ; : Pw . derations in which the curve Er is left out of account. But that ar the behaviour of mixtures for which more complicated phenomena occur, cannot be accounted for but by taking into consideration that a? can be — 0, seems also beyond doubt to me. az The approximate equation of state gives for this quantity the following value: lb? d°b : — MRT(Z\ wrr? pe ey. MRT de dz? dz? == + de elek (v—b)? v—b ee which I shall still somewhat simplify by assuming that 6 depends 2 linearly on «, and so = 0. We can easily derive from this form & J2 ‚4 W : : 5 that if —— can be == 0, this will be the case in a closed curve. At da? Yaa . dw. the boundaries of the v,7-diagram i 8 certainly positive. For «= 0 a : and 241 even infinitely great. Also for v= 6. And for v=o MRT | ——~, the minimum value of which is ey h is equal to iW 7 r Se 5 8g ic aly ry . 1 © 4 MRT. That, if only 7 is taken low enough, it can be negative, 2 . ak a . . . . . . at least if ze 18 positive, is also obvious. At exceedingly low value x J it reduces to of 7 it can take up a pretty large part of the v,x-diagram, which must especially be sought in the region of the small volumes. With rise of temperature this locus contracts, and at a certain maximum temperature for its existence, it reduces to a single point. So it is no longer found above a certain temperature. (To be continued). ( 639 ) Physics. “On the shape of the three-phase-line solid-liquid-vapour for a binary miature.” By Dr. Pa. Konnstamm. (Communicated by Prof. J. D. van per Waats.) Already for a considerable time I have been engaged in arranging Prof. Van per Waats’ thermodynamic lectures, and having arrived at the discussion of the three phase line solid-liquid-vapour, and the metastable and unstable equilibria solid-fluid which are in connection with it, I have formed on some points a different opinion from that laid down in the literature known to me on this subject.” It does not seem unprofitable to me to shortly discuss the points of deviation in this and the following communication. The first concerns the shape of the three phase line solid-liquid- vapour when the solid substance is one of the components, viz. the least volatile one. We find given for this that this line must always possess a pressure maximum ), and that it must also possess a temperature maximum’) when the solid substance, — as is usual, — melts with expansion of volume. The latter remark is the generalisation of a supposition, advanced by Van per Waats*) with respect to the line for ether and anthraquinone. These consider- ations, however, hold only for definite assumptions on the extent of the difference of volatility of the two components. This appears immediately from the differential equation of the three phase line given by VAN DER Waals‘) : Ly Ns — = Hs) dp x] i GAM he ca dil Ly Urs Us (Pi) aX in which 4, zand» denote resp. entropy, concentration and volume of the coexisting phases, the index v, / and s denoting that resp. the vapour, liquid and solid phase is meant. «x, does not occur, because we assume, that the solid phase is the first component itself so v, = 0. The pressure maximum will now occur in the line when the numerator, the temperature maximum when the denominator can become zero. Now ,—7; > %i—y, and Vy—V; > vji—v;; the two cases are therefore only possible when w, >, i.e. when the vapour is richer in the component which does not form the solid phase, 1) Baknuis Roozesoom. Die heterogenen Gleichgewichte Il. p. 331. 2) Smits. These Proc. VIII, p. 196; Zeitsch. phys. Ch. LIV, p. 498. 8) These Proc. VI, p. 248. t) Verslag Kon. Akademie V, p. 490. Ak ( 640 ) (for in the equation is put z;,==0) than the liquid. Or in other words, as we said above, the points sought can only present them- selves in the three phase line with the least volatile component as solid substance’). However, whether those points wi// occur, depends d Ly 5 . on the value which hm () will get. If this value may be put ml ARE ee so equal to the slope of the melting-point curve. So we must have both pressure and temperature maximum, at least when the solid substance expands on melting. This was the purport of the above cited remark of vaN DER WaaLs about ether and anthraquinone; if Ly : : ; 5 ‘ however (2) may not be put infinite, this conclusion is no longer Xl valid ; it then depends on the value which: r=0 Dy Ui Og (wi EE, Vs) Cl assumes for «=O whether there exists a temperature maximum or not; if the difference in volatility, so >, should not be so large, U] that this expression becomes negative at the limit, the maximum does not occur, even when vs > vj. The question whether such a maximum will occur in many systems, cannot be answered with certainty for the present. For this many data would be required, which we have not at our disposal as yet ; it is, however, possible to show the probability that only in very extreme cases the volability of the components will be so diversified, that a temperature maximum is to be expected. For this maximum to be just present, viz. in the triple point of the solid component, it is evidently required that : Vy Dijk 1 DL Vo Vs Now the first datum we should want, would be the variation of volume during melting. It seems, however, that only a few data have been collected for this; I have found some in WINKELMANN’s “Hand- 1) If has of course been tacitly assumed here, that there is no maximum vapour pressure; in that case the points in question could be found in both three phase lines. ( 641 ) buch” '), and in Baknurs Roozrgoom °) ; LANDOLT’s an tables do not give anything on this subject. The values indicated at the places mentioned confirm that the percentage of these expansions is not very considerable, which was a priori to be expected ; they amount for the highest cases to little more than 10°/, and for most substances they are considerably lower. So if we take 10°/, as basis, we shall find for by far the majority of the cases a too great, so for our proof a too unfavourable value. If we introduce this value, we get as condition (neglecting v by the side of wv): 3 Ul Vy So we must now try and get a rough estimation of the relation between liquid and vapour volume in the triple point. If at the triple point the vapour tension was of the order of an atmosphere, this ratio would be about of the order of magnitude 1000. Now, however, the vapour tension is always very considerably lower ; almost for every substance the melting point lies very considerably below the boiling point. If we now assume that the triple point lies at about */, 7%, we find the order of the vapour tension from the well known formula: With f=7 and T='/, Ty this gives log Ten If we put p, at 100 atms.*), p. becomes of the order of 0.1 atm. So we may safely say that in general v,/v, will be smaller than 0.0001. For a temperature maximum it is, therefore, necessary, that at least : SS TOR Pea id = = 70" or l. nat — —-11.5. Tl Ul Now according to a formula which has been repeatedly derived by vaN DER Waats‘), for low temperatures (a condition which in this case is certainly fulfilled) the equation : wee ee ad 1 aT, pea 9 l—az 2, ~~ \m Ty, dz b- da’ 1) II p. 612 2nd p. 775. eel p:'89. 5) In the table of Lanporr and Börnsrein only two substances occur, ammoniac and water which have a higher pe; the majority by far is considerably lower, particularly that of the little volatile substances which we have in view. 4) See e.g. These Proc. VII, p. 159. l ( 642 ) 7 holds, or for the limit, where == Sth) z zi dT: 1 db l i == > 1 Ep trent oe . « . . 2 es Ly Je dx b dx (2) It is clear that everything will depend on the first term here, because the second would not amount to more than —1 in the utmost case, i.e. when the 6 of the other component would be zero. Moreover it might even be possible that the second term was positive, it would hence decrease the value of the second member. The greatest difficulty for our calculation lies now in our igno- rance of with the variability of 7; with z, or more strictly in this that for this variability not one fixed ruie is to be given, because in every special case it will depend on the special properties of the mixture in question, viz. on the quantity a,,, a quantity which does not admit of being expressed *) in the characteristic quantities of the components, at least for the present. It is, therefore, certainly not permissible to try and derive results for all kinds of systems. But it is only our purpose to determine the course of 7), for those cases, in which the components differ exceedingly much in volatility, and for those cases it is perhaps not too inaccurate a supposition to assume for the present that the line which represents 7}, as function of «, does not deviate too much from a straight one. *) On this supposi- dy ee eg aaa tion then we may write En for k . AS now je 14, as Tr dx m bl we already supposed, —— ky must not descend considerably below 1) The equation of GALITZINE-BERTHELOT ds =, dy, which I rejected as general rule already on a former occasion on account of the properties of the mixture ether-choroform (These Proc. 1V, p. 159), can certainly not be accepted as such. Not only is it easy to mention other examples which are incompatible with this rule (see e.g. Quit, Thesis for the doctorate p. 44: Gerrits, Thesis for the doctorate, p. 68); but besides, — and perhaps this must be considered as a still more serious objection — by assuming this equation we wilfully break up the unity of the isopiestic figure (v. p. Waats, Proc. of this meeting p. 627) by pronouncing its middle region on the left of the asymptote to be impossible, whereas the left and right regions are considered as real. For if ay = Vajda itis never possible that da/dx = 0 for whatever system; and this takes exactly place in the middle region. I had overlooked this in the paper mentioned; Prof. vAN DER WAALS has since drawn my attention to it. The already mentioned system of Quint gives an da example of the occurrence of this case — =0; dj is there smaller than even the lac smallest of the two a’s. 2) Cf. vAN DER WAALs, These Proc. VIII, p. 272, ( 643 ) Ly : 0.9, that log. — may not become smaller than the required value 2] 11.5, or in other words, for the maximum in temperature to be reached, the critical temperature of one component must be about ten times as high as that of the other. A system, in which hydrogen occurs, will most likely show the temperature maximum when the other component has its critical point above 0° C., but already when the more volatile component is nitrogen or oxygen, we shall be more restricted in the choice of the other component. For then the latter must have its critical point at ‘about 1000° C. resp. 1250° C. If ether were the more volatile component, this temperature would almost amount to 4500° C. This conclusion is hardly affected when we put the temperature of the melting point not at '/,, but at */, of the critical temperature, as it really is for a number of substances whose critical temperature and melting temperature are known. It is true that this consi- Ly , but derably increases the second member of equation (2), and so ©} : By ML RE ‘ : in the same ratio — increases too, so that the quotient remains about vy unchanged. This is most easily seen when the condition on whicha temperature Maximum occurs, is written : wv Vy av —< 0.1 or log.— + log. vy — log. vi < log. 0.1. Ev Ly VI x] ; : Now for log. — we may introduce the value from the equation '): vy zy 1l—x#, ke € dT}. 1 zl log. ie l—azj xy T dz px de and write for log. v,: MRT i log. vy = log. = log. MRT + AF — 1) — log. Pk c so that the condition becomes: dT}, f Tyda So an increase of 7 will only affect the first term and the term log. MRT, and the logarithmic change of the latter will certainly amount to less than the change of the former. This now increases when 7’ becomes smaller, hence when at 7’= '/, 7), the inequality m 1 ae EC ee ee log. Pk — log. vl —f log. MRT i log. als 1) These Proc. VII, p. 559. ( 644 ) is not satisfied, this will certainly not be the case for 7’='/, Ty. Still, it would be too hazardous to assert that it has now been incontestably proved that e.g. for the system ether-anthraquinone no temperature maximum can occur. For we have had to make use of the supposition that 7), depends linearly on «, and though this supposition may possess some degree of probability for critical tem- peratures that differ much, it is just with substances which — as ether and anthraquinone-lie closer together, that there is some ground for expecting a deviation from the straight line. Only very few ex- perimental data are at our disposal. As such may e.g. be used the determinations on the increase of the plaitpoint temperature by addition of little volatile substances, made by Smits, CENTNERSZWER and Bicuner. For by means of the formula given by vaN per Waals’) ar dT, AS CATS ap Tde, Tide | 45 (Tyde 7 ppda (3) in which we need only pay regard to the principal terms (those dT. 7 from those directly Al with 7), we may calculate the value of ry’ measured. If we now calculate by the aid of the thus found dT si ut A value of Tide and the supposition of rectilinearity, 7, i.e. the Lda value of 7% for the admixed substance, we find the data collected in the following table. (P. 645). From this appears that the values calculated in this way at least for some substances, and particularly for anthraquinone according to the determination by Sirs, are not inconsiderably lower than double the melting point temperature. It may, therefore, be considered highly probable that these lines are convex seen from below, and so the 2 i absolute value of Td will be larger than might be expected from at the supposition of rectilinearity. With our imperfect knowledge of the further course of the plaitpoint line, and hence a fortiori of the line for 7; an estimation as to this will, naturally, remain very uncertain; but yet it seems to me that something about this may be ascertained in the following way. We have on the side of the ether: dT. 2 da db T .dz r=0 i ada th bdr be oi 1) These Proc. VII, p. 272 and 296. ( 645 ) | First Second FT. cal-| Double the Observer melting-point component component culated | temperature ————————— Anthraquinone | Ether SMITS 9320 1120° | rf SO, | CENTNERSZWER | 1032 1120 Resorcin = i 903 960 Camphor * EN 790 900 Naphthaline * 4. 770 700 5 CO, BiicHNER 640 700 | Paradichloro benzene 7 3 670 650 Paradibromo benzene - 7 690 720 Bromoform „ if 640 560 Orthochloronitro benz. as Fr 760 610 BücHNER'’s values have been borrowed from his thesis for the doctorate | (Amsterdam 1905); those of CENTNERSZWER from a table by van LAAR (These Proc. VIII, p. 151); that of Sirs has been calculated from his | determination: plaitpoint at 203° and’ 2 — 0.015, (These Proc. VII, p. 179). and so when introducing for a the quadratic and for 4 the linear function: (=) 20, = 26, : Tide) a, ¥ b, nie Now it will not be too hazardous an estimation, when — keeping in view that the formula for ether is C,H,,O and for anthraquinone C,,H,O, —, we put the size of the anthraquinone molecule at about two or three times that of the ether molecule; so 6, = 20, à 35. (4) } dT. If we introduce this value and the value of Tide’ calculated by the „de aid of equation (3), into equation (4), we obtain a value for a,,. Assuming that the value of 7% for anthraquinone is 2<560°—1120°, we can find an a, from the ratio of the critical temperatures ot ether and anthraquinone, and the a for ether; and with these quan- tities we can finally calculate the on the anthraquinone side da from : dT; 2a, (A) pe a, b, ( 646 ) A B N x > ; aT, Starting from 6,—= 26, we find in this way ( ) = (166: k Cv with 6, = 2.5: 0.65 and with 6,=36,: 0.64. The error which we committed in our choice of b,, will, therefore, bring about no con- siderable modification in the result; it would, indeed, be considerably modified if the critical point of anthraquinone should prove to lie considerably higher than 1120°. This is not in contradiction with our former remark that it is of little importance whether the reduced temperature is */, or '/, at the triple point; for this we started from the supposition of the linear dependence, whereas here we have abandoned this supposition, and calculate this dependence from the experimental data. So according to the course of reasoning followed here the a,, is given by the experiment, and the smaller value of m would now result in a higher value of a, at given b,,b, and a, If our estimation may be considered as not too inaccurate, we may conclude that the deviation from rectilinearity does increase the value of dT}; (=). but by no means in the degree which would be required to reach the critical value 0.9. (The value derived from the supposition of rectilinearity is 0,58). Though the foregoing calculations teach us hardly anything positive, they fix first of all our attention on the great desirability of more data concerning the values of the quantities « and 5 of very little volatile substances; for it appears again that the whole behaviour of all the systems in which such substances appear, is controlled by these quantities, and it would exactly be of great importance for the theory of mixtures, if its results could be tested by such cases where the properties of the two components differ strongly. It is true that it will not be easy to determine the critical point of such sub- stances in the usual way, but we should have gained already much if we could obtain an estimation of the critical temperature by calculation of the a and #6 from the deviations from the law of Boyre in rarefied gas state, so still some hundreds of degrees below the critical point. And further I think that after the foregoing I may be allowed to draw this conclusion, that the appearance of a temperature maximum in the three phase line, far from being thé general case, will be confined to mixtures of very exceptional nature. Tp oil Much more frequently than a temperature maximum will a pres- sure maximum occur. It appears from equation (1) that this will always be the case, when the expression: ( 647 ) Ly (no — Ns) — — (nt — 19) al may become negative. Now it is true that we cannot properly say that 1; — ns is a heat of sublimation and y; — 4s a latent heat of melting, because the 7’s do not refer to the same concentration, but we may say that 7,— ns is of the order of magnitude of a heat of sublimation, 4; — 4, of the order of a latent heat of melting. Or in other words 4, — 4, will be about 7 or 8 times 17 — ys. So in all & cases where (=) <7 the pressure maximum in the three phase Dl) 0 line will also fail. Here too the necessary data are wanting to ascertain whether there are many systems for which the ~ at the l triple point will descend to this amount. For, determinations of vapour tension or direct determinations of the required ratio have been nearly always carried out at considerably higher temperature *), and for the calculation by the aid of the just used formula the necessary data fail here too; besides, it would be doubtful whether the formula would be accurate enough, now that we have to deal with such small amounts. But — quite apart from the existence of mix- tures with minimum vapour pressure — the existence of a system like ether-chloroform ®) where on the chloroform side x, becomes almost equal to «,, already proves, that such systems exist. In any case to the scheme for the possible course of the two three phase lines in a binary system plotted by Baknuis RoozrBooMm in Fig. 108 of Vol. Il of his “Heterogene Gleichgewichte’, must be added types VII and VIII, characterized by a succession of sections, 1) Particularly when we notice that the ratio of 2» and x would have to be calculated from the formula: 1 dp Uy a ] dp Uyt] a a ON Sa pdx, x,(1—x,) p de; x |(1—a) and the value obtained will, therefore, strongly vary in consequence of a change of temperature of some ten degrees, which have generally an enormous influence per cent on the pressure in the neighbourhood of the triple point. 2) KOHNSTAMM and vAN-DALFSEN, These Proc. IV, p. 159. BAkHuIs RoozEBoom (le. I p. 41) deems it probable that also systems of gases with water and of water with many salts will show a similar shape. However, for such systems whose three phase line for the least volatile substance shows a pressure maximum, at least at temperatures that do not lie too far from the triple point, the shape of the p,a-line will have to deviate considerably from the line drawn there in Figs. 15 and 19, because from that shape would follow x, = 2. 44 Proceedings Royal Acad. Amsterdam. Vol. IX. ( 648 ) denoted by 1.7.4.5 and 1.7.8.5 in Roozssoow’s nomenclature 1). Type VI (see Fig. 1), is therefore distinguished from III in this that Fig Fig. 2. Fig. 3. Fig. 4. section 3 disappears; our Fig. 3 (lacking with RoozrBoom) takes its place. Type VIII (see Fig. 2) is distinguished from type V in this that instead of section 6 the section indicated in Fig. 4 appears between 8 and 5. Physics. — “On metastable and unstable equilibria solid-fluid.” By Dr. Pm. Konunstamm. (Communicated by Prof. J. D. van DER WAALS.) In a preeêding communication *) I discussed a point on which I could not agree with the existing literature on the equilibria solid- fluid. A second point which will prove to be allied to the preceding one, concerns the course of the curves which are to indicate the 1) Loc. cit. p. 392. 2) Proceeding of this meeting, p. 639. ( 649 ) metastable and unstable equilibria solid-fluid in the 7’, z-figures drawn up by vaN DER Waars *), and the w,r- and p, v-figures drawn up by Smits’). Van per Waars himself has already pointed out a defect in those figures *), viz. that the spinodal curve falls here within the connodal one, whereas in reality it falls far outside it at low tem- peratures; but it is not this that I have in view. Let us first take the p, x-figures. According to them the complica- tion which the binodal curve solid-fluid shows for temperatures below the triple-point, will disappear in this sense that at the triple point a new complication makes its appearance with three phase pressure, horizontal and vertical tangent, that then these two complications together give rise to the existence of a detached closed branch which contracts more and more, and at last disappears as isolated point. It is clear that in this way it is supposed that the complication can only disappear above the triple point, and not in the triple point itself, or in other words, that when the triple point is passed, always another three phase pressure is added to the existing one, and that these two more or less high, but always above the triple point pressure and the triple point temperature concur and disappear. Or expressed in another way still, it has been supposed in these figures that there is- always found a temperature maximum in the three phase line. In the light of the considerations of our preceding com- munication this supposition is by no means legitimate. But apart from this there rise serious objections against these views. First of all, if these views are held, it is impossible to see what the shape of the binodal curve solid-fluid must be when the solid substance is the more volatile component. Moreover all through the succession of the p, x-figures the binodal curve solid-fluid has only one point in common with the axis 2—0Q. Now it is, however, known, that for the com- ponents themselves, so for the concentrations «=O and «= 1 the p,T-diagram of fig. 1 holds (see the plate), i.e. at the triple point temperature there exists by the side of the triple point pressure C a second pressure of equilibrium solid-fluid (viz. of an unstable phase) C’, and above and below the triple point temperature these exist even two such pressures, one of which indicates metastable equilibrium, the other unstable equilibrium. But then the binodal curve solid-fluid for the mixture will not have to cut the axis of the component which becomes solid, once, but three times. And finally the p,«-figures of Smits and the 7’, x-figures of vaN per Waats 1) These Proc. VIII p. 193. 2) These Proc. VIII p.. 196; Spee, eit p.. 95: 44* ( 650 ) cannot be made to harmonize with the v, x-figures plotted by the former; for in these threefold intersection of the binodal with the rim does really occur before the detachment takes place (Compare in fig.6 of the said paper by Smits the line fed with f,c,e‚e, cf. between this a v,a-line must necessarily be found intersecting the rim in three points). Now that attention has once been drawn to these unstable and metastable equilibria, it seems desirable to remove these discrepancies. For this purpose the best thing is to start from the wv, z-figure. The general equation of coexistence of phases in the variables », x and 7 becomes in this case, if we now consider phase 2 as solid phase, 1 as fluid phase *): “yp op | ep dp ary RR i ze of dae t= de CaO) eae OAT apo zr | + (2 0 oa tt + dee fy + Oa Ff? a (=) PPO) so that we get for constant temperature : 0?) dw de Bo) tage (aorta du 0? 0? , Sa 5 — 0) + Oe (eee) In what follows we shall denote the numerator and the denomi- nator of this fraction by N and D. The geometrical meaning of D has already been given by vaN per Waats in his first paper on these subjects’): the locus D =O is the locus of the points of contact on the tangents drawn from the point for the solid substance to the isobars. It is easily shown that the locus N =O is the locus obtained by putting the q-lines i.e. the lines on C in this instead of the p-lines. So a double point or an isolated point, as they are assumed by Smits, tan only occur where the loci N= 0 and D=O intersect. As in such a point, as appears from the geometric meaning, the p- and the q-lines have the same tangent, and accordingly touch, such a point must also lie on the spinodal line ®). In perfect agreement 1) Cont. II p. 104. 2) These Proc. VI p. 233. 3) For from the equation of the spinodal curve dp 0 0? yp 0’ OP 3 Ovdw Ou? SS e= — 0 or = — = —— Ov? Ox? Ordv Op OT Ov? Ovda ( 654; 5 with this we easily obtain for the case that v, is not O or 1, the course of the loci mentioned indicated in fig.2. The dotted line denotes the concentration of the solid phase P; the lines A Q Band CQ D are the two branches of the spinodal curve, the two other lines wr . ae 0 joining A with B and C with D the branches of (5 )=0 When v/a now «=O becomes, it is evident that at this rim the line D—O dp BR must pass through the point where ee 0, and this point coincides, as is known, with the spinodal curve at the rim. The conclusion seems obvious that_the points Q and Q’, the points of intersection of the spinodal curve and ) =O have shifted towards the rim, and that, accordingly, the points of detachment and contraction from figs. 2—8 of Prof. Smits (loc. cit.) would have to lie at the rim. However, this conclusion would not be correct. For the inference that where the spinodal curve and D= 0 intersect, on account of the geometrical meaning of D=0O and N= 0, the latter must also intersect, does 0*?y not hold good at the rim. This is in connection with RES becoming Vv 0? ia, aes zero and — becoming infinite. [f we introduce the value MRT7/z, wv which the last quantity for «—O gets, then assumes the value: Op MRT ap j aan Ys = Wd af or la: ey — vs) — MRI and in general this expression will by no means be equal to zero Op in the points where ED = 0, as already appears from the simple OP — 3 : ‘ MRT consideration that there can be no connection between ——, uf—Us quantity which depends exclusively on the properties of the pure Op ge component and & in the maximum and minimum points of its v v isotherm, because this latter quantity will also have to depend on the properties of the second component. So the points Q and Q’ will certainly not lie at the rim, and in the points where D=0O follows: TT (dp\ Op Oy 0g (=) Ov? ss Orv @ dp dp dw 0g do a ve Ovda is Ou?” ey @) a é a ae i) ( 652 ) cuts the rim the binodal curve will simply have a tangent parallel to the v-axis. The shape which the different figures will assume, will now depend entirely on the fact whether such points Q and Q will also exist when the solid substance is one of the components and if so, where they lie. The best and most general way of solving these questions would be a full consideration of the different forms which the g-lines may present. As however the solution of the special question we are dealing with does not call for such a discussion, I believed to be justified in preferring another briefer mode of reasoning. For this purpose I point out first of all that it is easy to see. that at least in a special case such a point must exist also now. Let us imagine a plait, the plaitpoint of which has shifted so far to the side of the small volumes, that the tangent to the plait in the plaitpoint points towards the point indicating the solid state’). The plait touching the isobar in the plaitpoint, the plaitpoint lies evidently on the line D=O0 in this case’). But the plaitpoint lies also on the spinodal line, 2b so the point Q lies here in the plaitpoint, as neither De = 0, nor v 0 or = 0. We may conclude from this that in such like cases, so av those cases where the plaitpoint has been displaced still somewhat further or somewhat less far to the side of the small volumes, and perhaps in general when the difference in volatility between the two components is great, a branch of MN =O will pass through the figure, and that it will most likely have a point of intersection with the line D=0. A closer investigation of this supposition can, of course, only be given by the calculation. 1) The above was written before Prof. ONNES’ remarkable experiment (These Proc. VIII p 459) had called attention to “barotropic” plaitpoints. Now that the investigations started by this experiment have furnished the proof that plaitpoints can exist, in which the tangent runs // x-axis, the existence of plaitpoints as assumed in the text, in which the slope of the tangent need not even be so very smalì, has, of course been a fortiori proved. *) We may cursorily remark that it is therefore not correct to say in general that the line DO runs round the plait in the sense which VAN DER WAALS (These Proc. VIII p. 361) evidently attaches to this expression, i.e. that the point of intersection of the line DO with the binodal and spinodal curves would lie on either side of the plaitpoint. For if the plaitpoint should have moved still a little further to the side of the small volumes, the two points of intersection of D=0 with binodal and spinodal curves lie evidently on the vapour branch of these lines (the part of these lines between the plaitpoint and the point with the largest volume on the z-axis). ( 653 ) For this we shall start with the case that / increases and a decreases Op with increasing z, so that 7}, decreases strongly, and & is positive av v everywhere; and for the present we confine ourselves only to the solidification of the least volatile component, so «,—= 0. Let us write the value which MN gets at the rim by the aid of the value derived for 0 , from the equation of state, in the form: v/v MRT db 4/4, (ebde oo It is clear that this value will become negative for v=o, on the contrary positive for v="); so there will always have to be a point on the axis z—0, where N=0O. The value which MN assumes for «—1, is: Op ( MRT & a ve) AR 1—«z and this expression will, accordingly, be negative for «= 1 for all possible liquid volumes, and even negative infinite. From this follows that from the point of intersection of MN =O with the axis # — 0, the locus WV =O will run to smaller volumes. Now whether V=0O and DO will intersect in our figure depends on the place where N =O cuts the axis # — 0. In this we may distinguish three cases: 1. The point of intersection of N =0 and the axis lies at smaller volume than the points where D—O cuts the axis. Then no inter- section of MV = 0 and D=O will take place; the points Q and Q lie quite outside the axes c—1 and x= 0; 2. The point of intersection N =0 with the axis lies between the points of intersection of DO with it. Then the point of detachment does fall inside the figure, but not the point of contraction ; 3. The point of intersection of WV —O and the axis lies at larger (vs — vs) SMR Vijf lie Shiva (1) ') If we should object to putting v —=b, yet assuming that vs > v; , we shall in any case have to grant that there is nothing incompatible in the assumption that at a certain high pressure the volume in the solid state can be smaller than that in the liquid state, and that yet a great increase of pressure may be required to keep the substance ip the same volume after we have replaced some of the molecules by much larger ones (so (Si )=» Uy 2) As said, in every point of the line N =O the g-line passing through it, is directed to the point indicating the solid substance. Every g-line for infinite volume being // v-axis, and terminating in the point v =), it follows from the existence of the line N =O that every q-line cutting this locus, must at least possess one point of inflection. ( 654 ) volume. Then both the point of detachment and the point of contraction fall in the* figure. The consequences for the change of the v‚r-projection of the binodal curve with variation of temperature will probably be clear from the figures 8—5 without further elucidation in these three cases. With regard to the frequency with which the three cases occur, it is evident that the last case will occur only rarely, with exceptionally high values of db da _ (0p — and — —, or in general of {—- |. This case would be altogether da dx (32) = impossible, if we had to take the temperature of the triple point, and the volume which the saturated vapour then has, into account, for this amounts certainly to some thousands of times 6, and hence there will probably never be any question of an intersection of N=O with the branch DO holding for the large volumes at the triple point temperature. But for our case we have not to reckon with this temperature, but with the highest temperature at which the binodal curve solid-fluid has still three points in common with the axis x—Q0O, and this is evidently the temperature of point A in fig. 1. This, now, can probably lie very considerably above the triple point, and moreover — as we observed before — not the volume of the saturated vapour, but the much smaller one of the maximum of the isotherm must be introduced here. If we e.g. put the temperature of A so high that the maximum point of the isotherm lies at a volume 45, the expression will already become positive with ae or 6, = 4b, and »v, near b, (da/dz is negative). So the case of 3 is, indeed, possible, but it also appears that it will occur only in exceptional cases *). With none of these three cases do the 7’, z- and p, z-figures construed by var per Waars and Smits, agree. They agree in so far with that mentioned under 3, that the point of detachment and the point of contraction are assumed to fall within the figure. But it is at the same time clear from the va-figures, that a complication must begin ') It appears from what has been said here that the figures 6—9 are meant quite schematically, for though we have drawn several binodals solid-fluid which hold for different temperatures, we have left the loci N = 0 and D =O unchanged. This has, of course, been done to save space, for else we could not have repre- sented much more than one temperature in each figure without rendering the figures indistinct. But after what has been said it is clear that also the points Q and Q' move, and that it might e.g. very well happen that at lower tempera- tures the point Q’ is not yet present in the figure, and that it makes its appearance only at higher temperatures. The following figures, too, are meant schematically, and serve only to elucidate the properties mentioned in the text. ( 655 ) in the p,v- and 7,a-figures far below the triple point, viz. already at the temperature Bb of fig. 5, 1e. the temperature, at which in fig. 9 the new branch of the binodal curve (on the left side) makes its appearance in the figure. Let us first consider the p, z-lines. At the temperature mentioned (7) a new branch begins to form at the same height as the spinodal line, so far below the point of the stable coexistence. In the p, a-figure the point where this appears, is, in opposition with the v, x-figure, indeed a point where the tangent is indefinite ; for the equation: : dw dp dw 2 -— def, dvr” Oxf? dvs Oar holds for the former figure; the factor of dz, is zero on the spinodal line and the factor of dp on the line D=0O, which both pass through N . dp = (a: — a) ; dp, adel: the point considered; so Is there indefinite. The new branch extends Av more and more (fig. 6); its maximum continues to lie on the spino- dal curve, and the point with the vertical tangent on the line D=0. When the temperature of detachment in the w,z-figure (7) has been reached, the old branch and the new one unite (fig. 7), and separate again as figure 8 represents. At the triple point temperature (7) the middle one and the topmost one of the three points of inter- section with the axis coincide (in the final point of the double line vapour-liquid) (fig. 9); afterwards they exchange places. At still higher temperature the downmost point of intersection with the axis and that which has now become the middle one coincide; at this place there is again a point with indefinite tangent (7’,, the tempe- rature A of fig. 1) (fig. 10); at still higher temperature the binodal curve solid-fluid has got quite detached from the axis, and its downmost branch forms a closed curve, which contracts more and more, and at last disappears at the temperature of the isolated point of fig. 5. Here it is evidently essential that 7’, lies above 7’, and 7’, above 7,, according to the significance which they have in fig. 1; also T,, the point at which the detached branch disappears from the figure, must lie above 7’, the triple point, because in the triple point the binodal curve solid-liquid must still have two points in common with the rim (a little above it even three). But it is not essential that 7’, lies between 7’, and 7’; 7, might just as well lie above 7,. Then we get the succession: fig. 6, fig. 9a (triple point), fig. 10a. If now 7, lies below 7’,, there is confluence and section, and we get after fig. 10a fig. 11, and then Smits’ figs. 4 and 5 (loc. cit); if 7, lies also above T,, first the two lowest points of intersection of the binodal curve so!id- ( 656 ) liquid with the rim join, then they are detached from the rim, and we get, therefore, in this case, but only above 7’,, so above tempe- rature A of fig. 1, the continuous line drawn by Smits fig. 3 (loc. cit), which then passes into figs 4 and 5 (loc. cit.). The case mentioned under 2 that the point of contraction falls outside the figure may after all, be derived from the foregoing by putting 7, the temperature at which the detached branch disappears from the figure, below 7, the temperature at which it detaches itself from the rim. In our figures it has only this influence that the loop of figs. 9 and 10a cannot detach itself from the rim, as in fig. 10, and disappear as isolated point; but this loop contracts more and more at the rim and disappears there. In this case, too, 7’, can lie above 7’,, but of course, not above 7’. If 7, lies under 7;, we have the succession 6, 7, 8, 9 and disappearance of the loop in the rim; if 7, lies above 7, then: 6, 9a, 10a, 11, and disappearance of the loop in the rim. | The above case mentioned under 1, when also the point of detach- ment falls outside the v, a-figure, may be considered as the case that T, lies below 7, and 7, above 7. We have then the succession, the upper portion of fig. 6 (viz. without the downmost loop), figs. 12, 8, 9, after which the loop merges in the rim. Now in all the cases mentioned, except in the second subdivision of the case under 3 (so 7, above 7’), we meet still with two possibilities. Up to now we have assumed for those cases, that the triple point temperature is the highest temperature at which the two binodal curves intersect in the stable region, and that they have got detached above it. It is now, however, possible, that also in these cases the two binodals intersect twice at the triple point and above it. Then fig. 95, is put every- where for fig. 9, and then this is changed into fig. 11. We get then the following summary : Case under 1. Upper portion of 6, 12, 8, 9, disappearance of the loop in the rim 5 45 Wg seed baie GOOL ty se San EE Case under 2. Grits 5 0 a So pe ithe hee 6 9a. 100,081. + VEEN haere 62 78: (8b; ale. by soli, Scares ¢ AEN IE Case under 3. Gn OMO 0: disappearance of the loop in the fig. 6, 9a, 10a, 11, 4andbSurrs „ ie rende At Eon 6, 9a, 10a,-3, 4 anda murs. „ PEER SAE 6, 7, 8, 96, 11, 4and5 Smits The greatest chance to only one intersection with the binodal curve liquid-vapour presents, of course, as is best seen from the », v-figure, the case under 1, more particularly when in this case the line NV = 0 eats the axis at such small volumes, that it has no longer any point in common not only with the spinodal curve, but even with the binodal curve of the transverse plait. Only «ith a very exceptional course of the binodal curve of the transverse plait double intersection could take place in this case. On the other hand it will be highly probable that always when the line V=O cuts the binodal curve of the transverse plait (which will always have to take place in the cases under 2 and 3), also double intersection of the two binodals will be found. This shows at the same time the connection of this investigation with that of the preceding communication. For it appears now that the shape of the p‚z-lines holding for 1 with single intersection is, after all, by far the most frequently occurring, i.e. in almost all cases where no temperature maximum occurs in the three phase line ; for in this case the triple point temperature is the highest tempera- ture for which a three phase coexistence exists. For a complete survey I have also indicated in figs 183—16, how the binodal curve for the other solid phase gets detached from the trans- verse plait. This is only possible in one way, because here there cannot be intersection of the lines D—O and N=O. For v=1 or this binodal curve, and so the expression for 7’ at the rim becomes : Op MRT ne Ae har av vu av so always positive for both rims. The line N =O would therefore, have to become a closed curve, which on account of the shape of the g-lines may be considered as excluded *). In the 7z-lines double intersection will, of course, always occur above the triple point when the three phase line has a maximum pressure. For the rest nothing of interest is to be said of the 7'- lines; they have the same general course as the p‚z-lines given here, provided the figures are made to turn 180° round the z-axis, or in other words, provided a negative 7-axis is made of the p-axis. Then the points with vertical tangent lie here, of course, on the line W.,=0, instead of on the locus D =O; only at the rim they coin- 1) At least as long as the complications, which are in connection with the 2 existence of a locus —— = 0, do not present themselves. (See v. p. WAALs, Proc. da? of this meeting p. 637). 1 shall perhaps revert later on to the ee which are to be made in what precedes in consequence of this. ( 658 ) cide. If the pressure maximum of the three phase line should be found at higher pressure than the point A of fig. 1, we must, of course, have the case mentioned under 3, i.e. the point of contrac- tion must lie within the figure. It has been assumed in the above that throughout the region 0 se | pa os (32) = positive, and that a decreases with increasing 6. The case Ly that a increases with increasing 6 does not present any new points of view. If we have a system where a strongly increases, so that Op the critical temperature rises with 6 and (2 is negative, the ex- U)» pression Op\ MRT (v — vs) | 5 | + (ds — ef) —~ dx), xf(1—ae) is evidently always negative for a, = 0. And this is obvious, because this axis is now also that of the more volatile component; on the other hand the reversal of sign may now take place with the other axis. What happened on the left just now, will now take place on the right, and vice versa. It is only worthy of notice that now the line WV =O, if it exists, must intersect the axis #=1 in two points. For the expression MRT dh da/dz + MRT ve) ~ ((v—b)? de v* where db/de and da/dx are positive, becomes positive for v—b and v=o. From this follows that besides the just mentioned cases, another possibility may be found, i.e. that the point of contraction does fall within the figure, but not the point of detachment. For the p,a- and 7’, z-figures it makes only this difference that a loop formed in the way of fig. 12, (which always disappeared in the rim in the other cases) may now also disappear like the loop of ‘fig. 10 in a point within the figure. It is further clear, that in this case the point of contraction will much sooner fall within the figure db than in the preceding case. For according to formula (1) EE must at have an excessively high value for the expression to be able still to become positive with a volume v —= 105. If, however, da/de = 2a, — 2a,, =1.8a,'), then: 1) With the values for a and b of Lanpotr and Bérnstetn’s table 82 we find b: a about 12 for the highest value of xs about 250 for that of En if hydrogen is a 1 1 excluded, the values become resp. 8 and 40. So, whereas with exclusion of : Oa dz) hydrogen, pairs with a ratio > > 7 cannot occur, can reach the L ay value 39. | d if Dr. Ph, KOHNSTAMM On motastable and unstable equilibria solid-fluid da/da a MRT : =—18—=—1.8 — po v? v? m= B da/dx and so (v—vs) becomes of the order of magnitude 1.8 {MRT — v — p(v—v,)}. With this volume and the low temperature holding here the latter term is certainly a small fractiou of MRT, also MRT db aaa da ‚so that the expression becomes negative. 0 Nor does the case that (GE) may become zero in the examined & region, call for a further discussion, for it does not present any new Op 0a negative then positive (minimum critical temperature), we shall have on either side what in the first case took place on the left side (fig. 6—12); if da/de is first positive then negative (minimum of vapour pressure) we have on either side what happens on the right side in figs. 13 —16. ; da ) becomes zero in consequence of — v first being av points of view. If ( Nor does, in view of the foregoing, the occurrence of cases in which the plaitpoint curve meets the three phase line, offer any difficulty. It is only clear, that the two points where this meeting takes place, must lie below the point of detachment (double point of the binodal curve solid-fluid) both in pressure and in temperature. For when detach- ment has taken place, and so the binodal curve has split up into two branches, it seems no longer possible, when the v,x-figure constantly contracts and hence (ésf), has a negative value, that the three phase pressure coincides with a plaitpoint pressure *). But nothing indeed pleads against this conclusion. Only when we cling to the supposition that the point of detachment must always lie at the rim we are confronted by unsurmountable difficulties. For then the temperature and pressure of the point of detachment coincide with those of B (fig. 1), and this point, lying considerably below the triple point, lies certainly, at least in pressure, far below any plaitpoint. In conclusion we may remark that the cases where z, lies between 1 and 0, i.e. where the solid substance is a compound entirely or partially dissociated in the fluid state, may be derived in all their details from the v, x-figure (fig. 2) without any further difficulty. We get then at low temperatures Smits’ diagrams in the figures 4—-7 in his 1) Compare the figures referring to this in van per Waats, (These Proc. VI, . 237, VIII, p. 194 fig. (2) and Smirs (These Proc. VI, p. 491 and 495 and VIII, p. 200 (fig. 10). >, ( 660 ) paper: Contibution to the knowledge of the p, 2- and the p, 7-lines *,, at least when we take the maxima of pressure very much higher and the minima very much lower, so that on the left side the figure intersects itself twice. The detachment of the two binodal curves then takes place in a very intricate way by means of a series of modifications, which I shall, however, omit, with a view to the available space. So, for this I must refer to the lectures which I am arranging for publication as mentioned in the beginning of my preceding commu- nication, though certainly some time will elapse before they see the light. Physics. — “Contributions to the knowledge of the w-surface of Van per Waars. XIII On the conditions for the sinking and again rising of a gas phase in the liquid phase for binary mixtures. (continued). By Dr. W. H. Kersom. Communication N°. 96e from the Physical Laboratory at Leiden. (Communi- cated by Prof. H. KAMERLINGH ONNEs). (Communicated in the meeiing of January 26, 1907). § 6. Conditions for the occurrence of barotropic plaitpoints for mixtures with M,—=2M,, v%, =8v%,. Now that it had appeared in § 5 (These Proc. p. 510), that there exists a barotropic plaitpoint ’) on the assumptions mentioned there, first of all the occurrence of barotropic plaitpoints with M/M, = 2, vr, ‘or, = '/, was subjected to a closer investigation, partly also on account of the importance of these considerations for mixtures of He and H,’*). The barotropic plaitpoints given in table I respectively for the ratio of the critical ') These Proc. VIII, p. 200. 2) The proof that this barotropic plaitpoint really lies on the gas-liquid plait, is in connection with the discussion of the longitudinal plait. In a following Comm. by Prof. KAMERLINGH ONNEs and me on this latter subject, the treatment of which was postponed for the present as stated in Comm. N°. 964, the proof in question will be included. - 3) To enable us to judge in how far this last assumption is in accordance with what is known about mixtures of He and Hg, the following remark may follow hére, in the name of Prof. KAMERLINGH Onnes too, (cf. Comm. N° 965 § 4, Dec. ‘06 p. 506) on 6 (cf. van peR Waats, These Proc. Jan. ’07, p. 528) and a for helium: It proved in the preliminary experiment described in Comm. N°. 96a that on analysis the liquid phase contained at least (some He has evaporated from it during the drawing off of the liquid phase) about 3 °/) He, the gas phase at least (a very small quantity of liquid has been drawn along with the gas phase being blown off) about 21°/, Hy (estimations of. the corrections which for the reasons mentioned ought to be applied to the results of the analysis make it probable that they will not considerably influence the results derived here [added in the English translation]). Let us put the density of liquid hydrogen boiling under ( 661 ) temperatures of the components given there are found in the way explained in § 5. atmospheric pressure according to DewAr (Roy. Institution Weekly Evening Meetings), 25 March ’04) at 0.070, and let us derive the coefficient of compressibility according to the principle of the corresponding states, e.g. from that of pentane at 20° G,, then the density at 40 atms. is 0,072. If we calculate the increase of density in consequence of the solution of helium from van per Waars’ equation of state for : : 1 ; a binary mixture by putting DM He = > bMHz for this correction term, we get for the density of the liquid phase at the p and 7 mentioned if it contained 3 °/, He : 0.077. The gas phase will have the same density at about the p and 7’ mentioned (cf. Comm. N°. 96a, Nov. ’06, p. 460). The theoretical density (AvogApRo-BoyLe-Gay-Lussac) at T=20° and p= 40 atms. = 0.0885. If we assume Van per WaALs’ equation of state with a, and b, for constant « not dependent on v and 7, to hold for this gas phase, bx ar 1 v—br ET 2 = 0.00042 (Konnstamm, LANDOLT-BöRNSTEIN-Mevyer- it follows with the above given value of the density that = 0.15. For daz =, =0 with a 44, HOFFER’S Physik. Chem. Tabellen), and with vj 0.0021, putting 2 = 0.80 for the gas phase, we should obtain: by ae 0.21 v,,= 0.00044, We should then, if a rom Ey yg + Poos): St Doo et Do, 7, = 0.00033 =5 gb, 7 P11 ag = 9-00088, Kounstamm |. c.). If we wish to assume positive values for a and 433 (cf. Comm. N°. 96a, p. 460), we should have to put b, usb fer T = 20°; if we we may put D assumed that the gas phase contained 15%, He we should derive from the above mentioned experiment for positive values of aj, and dg: Doom > 0.31 Dum: These results harmonize very well with what may be derived about byype at O° C.; the ratio of the refracting powers (RayLEIGH) gives: Doo = 9.31 8,, w while the ratio of the coefficients of viscosity and also that of the coefficients of the conduction of heat lead to a greater value for yyy, (about 1, bij). If we take bas 44/0), 47 = 2, we should obtain from the above given considerations (putting dio Varnm doom): Goals = ars SO that Type = about 0.35°. This renders a value for the critical temperature of He <0.5° probable. This conclusion would not hold if bx3s for z =0.8 were considerably greater than follows from the hypothesis that br varies linearly with x. This however is according to the experiments of Kugenen, Keesom and BRINKMAN on mixtures of CH,Cl — CO, and CO, — Og, not to be expected. The experiments of VERSCHAFFELT on mixtures of CO; — H, would admit the possibility, but give no indication for the probability of it. [Added in the English translation]. So though probably %2:/,, for mixtures of He and Hy is larger, yet we shall here retain the supposition made in § 5 on bae/5,, with which the calculations were started, because the accurate amount is not yet known to us, and we only wish to give here an example for discussion; moreover the course of the y-sur- face will not be considerably modified by this difference in any essential respect. ( 662 ) TABLE I. | Barotropic plaitpoints at M/M, = */,, vz, [or = */s- | et ee AK! vl | TE |P opt Pry 0.0002 06 0.3957 1.005 | 4,805 | 0.0210 0:63 0.3481 0.934 4.772 4/121 2/3 1/3 AIA /484 576/121 0.0604 0.7 0.3048 0.867 4.158 0.1044 | 0.75 0 2636 0.800 4.780 0.4472 08 0.2934 0.726 4.814 0.1842 | 0.85 0.1833 0.638 | 4.800 0.2106 0.9 0.1494 | 0.521 4.567 0.2176 | 0.925 04212 | 0.444 4.202 0.2199 | 0.94 oost | 0.387 3.751 0.2182 | 0.95 0.0991 | 0.348 3,982 0.2148 | 0.965 | 0.0851 | 0.266 2.107 0.2106 | 0975 0.0752 | 0.204 0 687 02040 | 0985 | 0.064 0.430 — 1.927 01996 | 0.99 0.0585 0.078 | — 5.491 0.1960 | 0.995 0.0518 0 033 ~ 9.793 0.1956 | 0.9965 | 0.045 | 0.028 —10.396 | 0.1964 0.9975 0.0478 | 0.019 —12.086 4/4 1 ads ORD — 97 For so far as the assumed suppositions hold, the barotropic plait- points given in the table have only physical significance if 7',,, does not become so low that solid phases make their influence felt (ef. Comm. N°. 965, § 5 6), and if moreover the portion of the p-surface in the neighbourhood of the plait-peint is not covered by a portion of the derived surface indicating more stable equilibria (as e.g. will be the case for negative pressures). In how far the indicated baro- tropic plaitpoints will belong to the gas-liquid-plait will be more fully treated in a following communication (ef. footnote 2, p. 660). In the first place it follows from table I that with the assumed ( 663 ) ratios of the molecular weights and of the critical volumes, baro- tropic plaitpoints prove only possible (quite apart from the question whether they are physically realisable, and whether they belong to i the gas-liquid plait) for 77/7, <a The barotropic plaitpoint for the ratio 7%,/7)., = 0.0002 is a plaitpoint for a mixture, one component of which is a gas almost without cohesion (Comm. No. 965 $ 1). A further consideration of it would lead us again to the region of the longitudinal plait. The conditions relating to barotropic plaitpoints for «,,; near 1 furnish a contribution to the knowledge of van per WAALS’ y-surface for binary mixtures with a small proportion of one of the components '). We find for 7;,/7;, near */,, putting x); 1— &: PIT 1 1 | a ala : : a — — ¢/3 “13 — —— Pt! oc ee Ol it LS i 5 Sh + Tk ié 0 EE It is seen from the series of the ratios 71/7, in table I, that in this a maximum and a minimum occur, respectively for about T,,/T%, = 0.219 and 0.196. From the formulae derived for Vopl near 1, a minimum and a maximum for u is found, and hence for Til Th, respectively at ay,; = 0.9968 and 0.969. That the latter is in reality found at «,;=0.94 is due to following terms in the development. For 77/71, < 0.196 or 0.219 < 7;./7., < 0.25 one barotropic plaitpoint is found, for 0.196 < 77/7, < 0.219 three. In connec- tion with Comm. No. 965 $ 2 (Dec. ’06 p. 502, cf. also this Comm. $ 1, Dee. 06, p. 508) it follows also from this that for the mixtures considered here at lower temperature the longitudinal plait makes its influence felt. The experiment described in Comm. N°. 96a proved that for mixtures of He and H, at — 253°, ie. about 7= 0.65 Tin, a barotropic tangent chord is found on the w-surface. If at that temperature only one barotropic tangent chord occurs, this will point to this (Comm. N°. 965 p. 504) that for the mixtures of these sub- stances 7,1; > 0.65 74,, and therefore according to this table Tite < 0.18 Ti, while the found considerable difference in con- centration between the gas and the liquid phase (see Footnote 3, 1) Cf. Comm. N°. 75 (Dec. 701), NO 79 (April ’02), NO. 81 (Oct. ’02), Suppl. N°. 6 (May, June ’03). 45 Proceedings Royal Acad. Amsterdam. Vol. IX. ( 664 ) p. 660) indicates, that 7,7 would have to lie still pretty much higher, and therefore Tre pretty much lower (probably < about 4°)’). Of this result we availed ourselves in the treatment of the estimation of the critical temperature of He in Comm. N°. 96%. (To be continued). Physics. — “Contributions to the knowledge of the w-surface of VAN DER Waars. XIV. Graphical deduction of the results of KUENEN’s experiments on mixtures of ethane and nitrous oxide.” Supplement 14 to the Communications from the Physical Laboratory of Leiden. By Prof. H. KAMERLINGH Onnes and Miss T. C. Joures. (Communicated in the meeting of Januari 26, 1907). § 1. Introduction. In what follows we have endeavoured to derive quantitatively by first approximation the behaviour of the mixtures of N,O and C,H, (mixtures of the II type ’)), which has become known through KvENEN’s experiments *), by the aid of VAN DER Waars’ free-energy surface. The w-surfaces construed for this purpose (see plate I) are the counterparts of those construed in Comm. N°. 59 (These Proc. Sept. 1900) and Comm. N°. 64%) for the derivation of the results of KurNeN’s and HarrMan’s experiments on mixtures of CO, and CH,Cl (mixtures of the I type). In the graphical treat- ment °) of our problem we have chiefly followed the method given in Comm. N°. 59, where the critical temperature and pressure of some mixtures were borrowed from KUENEN’s determinations, and then the results of another group of experiments — those referring to the conditions of coexistence of two phases at a certain tempera- ture — were deduced by the aid of van DER Waats’ theory. KUENEN’s results for N,O and C,H, are principally laid down in 1) If bo2/s,, is taken larger than 1/, (Cf. Footnote 3 p. 660) this supposition too makes the upmost limit for 7, on the said supposition smaller. This is seen when we compare with table I that we obtain 7, / ad =0.679 for baat, = 1/4 . ~ . ‘1 with Dial Peete *) HARTMAN, Leiden Comm. Suppl. no. 3, p. 11. 3) Kuenen, Leiden Comm. no. 16, Phil. Mag. 40, p. 173, 1895, cf. also Kamer- LINGH Onnes and Zaxrzewski, Leiden Comm. Suppl. no. 8. (These Proc. Sept. 1904). It is remarkable that the possibility of this case was foreseen by van DER W aars, Contin. Il, p. 49 [added in the -English translation]. 4) Arch. Néerl. Serie IL, Tome V, p. 636. 5) Only graphical solutions for definite cases are here possible. (Cf. Suppl. 8, These Proc. Sept. 1904. § 1). ( 665 ) four figures'); one of them gives the critical quantities from which we shall start in our deduction, and the border curves for mixtures of different concentration; the three others, which represent the zv- projection of the connode with the connodal or tangent chords at the temperatures 20°C, 25°C, 26°C, show the contraction of the transverse plait with rise of temperature, and finally its splitting up into two plaits. We have thought that we could obtain a better comparison of observation and calculation, when representing the observations by the ev-figures for 5°C., 20°C. and 26° C., and the p7-figure, instead of by the zv-figures at the before mentioned temperatures and the pT-tigure. § 2. Basis of the calculation. Law of the corresponding states and reduced equation of state. We start (cf. Suppl. N°. 8, These Proce. Sept. 1904 § 1) from the supposition that the law of the corresponding states — at least within the region of the observations — holds as well for C,H, and N,O as for their mixtures. As reduced equation of state we chose equation V.s.1 of Comm. N°. 74?) p. 12 For a region of reduced temperature and pressure which incloses the region which corresponds to that of the observations under in- vestigation, this equation is as closely as possible adjusted to CO, which in thermical properties has much in common with N,O, and there is no reason to suppose that this will not be the case with C,H,. In the application we are, however, confronted by this difficulty, that V. s. 1 deviates most strongly from the observations on CO, exactly in the neighbourhood of the critical state. (Cf. Comm. N°. 74 and later Kersom, Comm. N°. 88). If from V. s. 1°) the point is derived, for which ee ’ dp dp — =.0 and — dv Ov? we find t—1.010595, Av —=1.0407.10-% p= 1,06566. Krrsom’s observations, Comm. N°. 88, give for the critical volume, when it is sought by the application of the law of the rectilinear diameter, for CO, vy, = 0.00418 and for i= t= 1.0027.10-3. . . (1) k 1) Where it was necessary, Kuenen’s figures have heen rectified in accordance with the results of observation given by him. 2) Arch. Néerl. Vol. Jubil. Bosscha. Serie II, tome 6, 1901. 3) In the calculations 7’ is put 273°,04 for the freezing point of water, because V. s. 1 was calculated with this value. 45* ( 666 ) We find then: fx == 1.010595 instead OE fd D= 10579 5 , Set pr = 1.06566 Ps tae The isotherms from which V. s. 1 has been derived by the com- putation of the virial coefficients B, € etc. (See comm. N°. 71, These Proc. June 1901), indicate therefore, by means of interpolation according to this mode of calculation, a critical state, which, drawn in the pv-diagram, has shifted with respect to that which was found by immediate observation; the critical temperature according to V. s. 1 is namely t,7% when 7%, is the observed critical temperature. So are also the values found for Pmaz., Puig and Yap, at t by the application of Maxwe..’s criterion, different from those which we should find when dividing Pax. by pr Viig. and Uap, by vp. The deviations are of the same order as the deviations of the substances inter se, when they are compared by the law of the corresponding states. At t= 0,9 they are about zero, but they increase as we approach the critical state, so that the deviations agree with a gradual transformation of the net of isotherms. The following table gives a survey of the deviations in the corresponding values. Column A refers to V.s.1 and implicitly to pr, Tr of CO, Ss B,, _,, the observations „00 EE) C EE) EE) ” ” ” N,O. EE ED | Somer ees Ed OA CE 1: 4 te | —_— 0.975 0.826 0:975 | 0.84 0.975 0.854 0.950 | 0.695 0.950 | 0.709 0.950 | 0.720 0.925 | 0.589 0.995 | 0.599 1.0106 1.066 | 0.900 | 0.490 0.900 | 0.494 0.900 0.486 In the neighbourhood of the critical temperature the phenomena are governed by the difference of the temperature of observation and the critical temperature, 7’ — 7;,; for this reason we have chosen for the detailed model of 26° such a temperature 7” for the com- ( 667 ) parison of the observations at that temperature 7’ with the result of computation that T,— T=t.%,—- T’. At the general survey for 26° we have applied to dk a cor- rection = At, so that Af—0.01060, when 9=10(t—0,9) for 0<6<1, whereas At=O for all other values of 4. The correction Attot was accompanied by a correction Av to v and Ap to p, so that Ay — 0.06574 and Av — 0.0379 6, which together represent a regular increase of the corrections from t= 0,9 to the critical state. For the detailed model of the y-surface for 26°, on which only the part from «0,35 to «0,65 was represented, we used everywhere the same correction viz. A t=0.0106, Ap = 0,0657, Av = 0,0379. § 3. Critical quantities for the mixtures. KurNeN has determined the plaitpoints 7, plas Ppla 5 Upla > for some mixtures with the molecular concentration 2. Pe, Pra, Ure, the critical points of contact, and Thx Pkx» Vix the critical states of the mixtures taken as homogeneous differ so little ) from these values, that this difference may be dis- regarded for our purpose, and so they are also known for the mixtures investigated by KUENEN. ry th Fat x = = apa 4 = IN FS id. a ae | + ede pe | | \ Ge 5 En — +— Al -— hen — Se | [eN ie | \ 6 as Set + — | Ne ee | \ sm + + DS te sot = ES | =" | a i. se + —}. + + —— | 5 A EE — + + ts#—- +—-+ — —— = e ASF on Ay os oye pn RT pal 0 as “a FR. Jig. 5. | | | Kas WENNEN + + + = Goor + —+ ey = 1 1) This has been fully treated by van per Waats, Cont, H.a§. AL, ( 668 ) The critical quantities for the other mixtures were found by gra- phical interpolation. Fig. 1 gives per, fig. 2 Tr, as function of z; in fig. 3 vj, has been calculated from py, and Tix, by tbe aid of 2 (see formula (1)); the vz, observed by Kurnen have been indicated there. By carrying out the construction for the connode by the aid of the and yy — # = —v a curves (see Comm. N°. 59a')), which the models for those different temperatures yield, we may derive vj, — Upc Pke — Ppl.x« in first approximation (see Comm. N°. 59a). Applying these corrections we should then have to repeat the calculation from the beginning, to obtain more accurate values for vj, Pex. We have confined ourselves to a first approximation in all our constructions, as also a further correction of the equation of state V.s. 1., which can cancel the deviations mentioned in § 2, has not been applied and we were the more justified in this, as these latter deviations are larger than those we have now in view. 0 PO § 4. Construction of the w-surfaces. From the equation of state V.s.1 we find immediately the reduced w,-7 curves, from which are then deduced the ordinary y-curves accor- ding to Suppl. N°. 8,§ 4; or the ordinary virial coefficients, whic hare then used for the calculation of yw according to Comm. N°. 59. For the construction (cf. Comm. 59) use was made of: at 35° y, =wt+ 0107+ 250 0 WW + 0,10 2 + 36,5 v ua wv, = w+ 0,24 x + 57,3 v, while a suitable constant was subtracted from every wp. Here v is expressed in the theoretical normal volume, just as in the diagrams. From the w,7 curves (cf. fig. 1, pl. ID) the w‚r curves (see fig. 2, pl. II) and the pr curves were graphically derived. The models for y were construed on a scale 5 times larger than the diagrams on plc Tl,pl. HI and ypleeiy: § 5. Determination of the coexisting phases. Applied was both the construction by rolling a glass plate on the 1) In giving the figure 3 in Comm. N°, 59a for this construction it was stated that this figure was very imperfect. It appears now that the loop ought to contain two cusps. We found out the error by the aid of the general propertes of the substitution-curves treated by van per Waars (Comp. Proceedings of this meeting). This error shows the more how necessary it is that graphical solutions are controlled by such general properties as van per Waats is now publishing. | Added in the English translation]. ( 669 ) model, which yields the connode and the tangent chords, and the simplified construction in the plane given in Suppl. N°. 8, § 7, to which a small correction was applied. After viz., a provisional connode, that of the mixtures taken as homogeneous, has been found by tracing curves of double contact to the y,7-curves, and by determining conjugate points 4 on the gas branch at some points a ” z 0 ) on the liquid branch of that connode, so that every time ~ is the U same for the two conjugate points, the lines which join every two of these points a and 5, are produced outside the provisional connode, till they cut the isobars which pass through a, in points c, which together represent the required gasbranch of the connode; c and a are then considered as conjugate points. In the y-surface at 5°C. the two constructions yielded fairly well corresponding results, both with regard to the chords and to the connode itself, as appears from pl. II fig. 4, where — — — — denotes the connode and the connodal tangent-chords found by rolling a glass-plate on the model, ee those found by means of the just mentioned construction. That the simplified construction, which was more particularly plotted for equilibria far below the critical temperature (see Suppl. N°. 8, § 7). still leads to our end, is probably due to the fact, that we have here to deal with a mixture of the II type. With the w-surface for 20° the slight depth of the plait rendered it necessary, to considerably diminish the longitudinal scale for the v-coordinate of the model. This compression (ef. pl. I, fig. 2) rendered the plait sufficiently clear to determine the connode and the place of the connodal tangent chords by roliing a glass plate. By means of the simplified construction the connode was still to be obtained, but the determination of the tangent-chords became uncertain. With the y-surface of 26° the depth of the plait (here split into two) becomes so exceedingly slight, that it does not appear but with a computation with 7 decimals, and even then it manifests itself almost quite in the two last decimals. Hence it is not possible to model a y-suriace (we mean a surface derived from the w-surface, on which the coexisting phases are still to be found by rolling a plane}, on which this plait is visible, nor is it of any avail to confine ourselves to a small part of the surface, because the curvature of the w,7- curves is very strong exactly there where something important might be shown. The determination of the connode and the connodal tangent chords by construction according to $8 of Comm. N°. 59a, which can always be carried out provided enough decimals are worked with, remained still uncertain up to 7 decimals, so that we have ( 670 ) not pursued it any further. Thus the represented part of the w-surface for 26° from #=0.35 to 70,65 and from v=0.0038 to v=0.0070, has been given by us chiefly to demonstrate how exceedingly small the influences must be on which a plait depends, and how much care is required to determine a plait experimentally which is not at all to be seen on the surface. The curves drawn on the surface, which relate to the plait, were found by indirect ways, partly by construction, partly by calculation. To facilitate a comparison of the models inter se the region of « and v, on which the model for 20° and that for 26° extends, has been indicated on the model for 5°, on the model for 20° that for 26°. § 6. Further remarks on the different models and drawings obtained by construction. a. The w-surface for 5°. The model, pl. I, fig. 1, and the drawings pl. II, figs. 1, 2 and 3 show curves of equal concentration, y',7, equal volume and equal pressure, the connode and the connodal ! 3 tangent chords. As— — = p—25, some pressures are represented On by negative slopes on the stable part of the w’-surface, in consequence of which the character of this w’-surface does not in this respect imme- diately express that of the y-surface, where all the slopes are positive. A connodal tangent-chord, near the concentration with maximum pressure, almost touches the y-line. With the concentration of maximum pressure this would be just the case. Just as the connodal tangent chords the isobars are traced in the projection on the x v-plane (Pl. Il, fig. 3) in full lines, the eonnode is denpted ye 2) 2 For the isobars*) we may note several peculiarities, to which VAN DER Waats has drawn attention in his theory of ternary systems *). The isobar which touches the connode on the liquid and vapour side, belongs to the pressure p = 36,6, which is found for the mixture which when behaving as a simple substance should: have a maximum coexistence pressure. The pressure curve x determines the transition between the continuous isobars (taking the region outside the drawing into consideration) and those split up into two branches. The parts of the continuous isobars which point to P, have each a point of inflection on either 0? 0? i =) and — = 0 Ovda Ov side of the top. The shape of the curves 1) Cf. the sketch by Harrman, Leiden Comm., Suppl n°. 3, pl. IL, fig. 5. 2) These Proc. March 1902, p. 540. ( 671 ) is as has been indicated by vaN per Waats '). The points of intersec- tion of these two curves are the centre Q of the isobars and the double point of the pressure curve a, P. b. The y'-surface for 20°. Fig. 1, pl. UL denotes the y,7-curves and the connode. Fig 2, pl. Ii the ,7-curves and the connode. Fig. 3 gives the projection on the wv-plane of the connode, of the tangent chords and of some isobars. The connode is denoted by __._. Pl. I fig. 2 gives a representation of the model. c. The w'- surface ‘for 26°. Hig, 4 pl. IV gives the y,7-curves, fig. 2 pl. IV gives the critical states, A, and A, the isobars and the con- nodes for the mixtures which are taken as homogeneous, and whose gas branch as well as whose liquid branch is almost a straight line. Though in the ealeulations (see § 2) the plaitpoint #7, and the criti- cal point of the homogeneous mixture zz, have been considered as coinciding, a distance. has now been given between these points which has been fixed by estimation ®). The dotted parabola has been taken from VerrscHAFFELT’s calculation, Suppl. N°. 7, p. 7, though properly speaking it holds only for the case that the maximum pressure falls in P, or P,; the produced connode denotes the probable course of this part by approximation. Pl. I, fig. 3 gives a repre- sentation of the model. All this refers to a small region of w and»; fig. 3 pl. IV, however, indicates by ._.__. the connode according to the construction for the mixtures taken as homogeneous all over the width of the w'-surface. The square drawn denotes the extension of the just treated part of the y’-surface. d. The contraction and the subsequent splitting up of the plait appears from fig. 4 Pl. III, where the «v-projections of the eonnode and some connodal tangent chords of the three models have been drawn on the same scale after the wv-figures for 5°, 20° and 26° mentioned under a bc. 1) Prof. van per Waats was so kind as to draw our attention to a property which might also have been represented in the figure, when also the curve for Ow ae? aan had been drawn, viz. that the minimum volume in the vapour branch, v' Ow 3 and the maximum volume in the liquid branch lie on the curve fi = 0 which Òvde 2 . . Us . . has a course similar to that of the curve Pee Med more particularly it has the vda same asymptotes, and it deviates from it only in this, that with greater density the curve passes over larger volumes. 2) Here the representation of the plait must come into conflict with the theory or with the simplification introduced at the basis of the calculation. With a view to the illustration of the theory by figures the latter has been chosen. ( 672 ) §7 Comparison of the construction with the observation. On the whole this is very satisfactory, taking the degree of approximation into consideration. a. In pl. II fig. 4 the diagram for the plait at 5° indicating Kugrnen’s observations, has been drawn in full lines. The figure contains at the same time that obtained by construction. The single observations have been denoted by [-| (see § 6a). Besides the con- struction with the model indicated by — — — — and by \-/, also the simplified constructions in the plane indicated by —_._.— and by ©, the outermost of which refers to the less simplified con- struction, represent the character and also the numerical values satis- factorily. b. In pl. Ill, fig. 3 the figure representing KUENEN’s observations for the plait of 20°, has been indicated by — — — —. The figure contains at the same time the __.__.__., obtained by construction on the model (see $ 65). The correspondence at «= 0.3 is the worst, which is no doubt in connection with this, that here we have already got very near the critical temperature, and that strictly speaking, different values should be assigned to 7” (see § 2) for all wv, and corresponding Av and Ap should have been taken into consideration. c. In pl. IV fig. 3 the figure representing KUENEN’s observations, have been indicated by full lines; the figure contains also the figure derived in §6c denoted by —._.— curves. d. Plate IV fig. 4 and 5 may serve for a comparison of KUENEN’S pl-figure (fig. 4) with that derived by construction (fig. 5). In accordance with the remark on 7” in $2, we have proceeded for 26° as follows : For 5° and 20° the values of p and 7’ have simply been taken from the construction with the model, mentioned under a and 5. Then we marked @) the p's and 7”s, obtained by multiplying KuENEN’s pr and Tj, by pr and fz (see § 2); B) for the different values of 7 the values of 7” and of 6 for the temperature of 26° have been calculated, and then Af and Ap determined by the aid of this 6 according to § 2; the values of p and 7’ corrected in this way have been denoted by +--+ + and joined by —_.—.—.— curves with the points men- tioned under a). The full and the dotted curves give the corrected values. Between the parts where we started from the critical temperature, and the p7-lines derived fromthe models of 5° and 20° a space has been left open. H. KAMERLINGH ONNES and Miss T. C. JOLLES Contributions to the knowledge of the J-surface of va W XIV. Graphical deduction of the results of Kurne experiments on mixtures of ethane and nitrous oxide Plate I. P, dings Royal Acad. Amsterdam. Vol. IX AN H KAMERLINGH ONNES and Miss T. C. JOLLES. Contributions to the knowledge of the v-surface of van per Waats. XIV. Graphical deduction of the results of Kuenen’s experiments on mixtures of ethane and nitroys oxide. ——= ee en = == eh | 1 Wit if Wt = 7 = = =| X 0 zap VOS 2025 XaG43 xe0,55 refe MOP Aad Tig 2 v vipa — il | = ee. jij Si med === : == = se ar en = nd Fig 5 Proceedings Royal Acad. Amsterdam. Vol. IX. Plate IL. v Da ial is H | | | | I ij It i i Pst | ne Pets aD NE B Tiga. tos H. KAMERLINGH ONNES and Miss T. C. JOLLES, Contributions to the knowledge of the J-surface of van pen WaAats . XIV. Graphical deduction of the results of Kuexen's experiments on mixtures of ethane and nitrous oxide, Plate III. en ou en 7 ope Proceedings Royal Acad. Amsterdam. Vol. IX. H. KAMERLINGH ONNES and Miss T C. JOLLES, Contributions to the knowledge of the y-surface of van pen Waars. XIV. Graphical deduction of the results of Kuenen’s experiments on mixtures of ethane and nitrous oxide. Plate IV. Fig. 5. Proceedings Royal Acad. Amsterdam. Vol. IX. ( 673 ) Pathology. — “On the Origin of Pulmonary Anthracosis.” By P. NieuweNHvijse. (From the Pathological Institute in Utrecht). (Communicated by Prof. C. H. H. Spronck.) (Communicated in the meeting of January 26, 1907). As is known, von BruriNG and CaLmerre oppose the doctrine accord- ing to which the pulmonary tuberculosis among mankind proceeds in most cases from inhalation or aspiration of tuberclebacilli. They presume the tractus intestinalis to be the porte d’entrée of the virus. In connection with this new hypothesis VANSTEENBERGHE and GRIskz') have made some experiments at the end of 1905 in CaALMeTTE’S laboratory about the origin of lung-anthracosis. They mixed the food of full-grown cavies with soot, Indian ink or carmine and made the animals eat a large quantity of this. After 24 hours already they found resp. black and red spots in the lungs especially in the upperlobes and along the edge of the underlobes. VANSTEENBERGHE and Geisrz concluded from these results that the fine parts, taken up in the intestines, pass through the mesenteric glands and thoracic duct and after having reached the blood in this way, they are caught by the lungs. According to their conclusion the carbon particles suspended in the atmosphere would not be inhaled, but swallowed, thus reaching the lungs via the intestines. The theory of the intestinal origin of the pulmo- nary anthracosis was propounded half a century ago by VILLARET®); it had however met with little success, and after the careful resear- ches made by Arnonp*) on the inhalation of fine particles it was totaliy forgotten. Whereas VANSTEENBERGHE and Grisez tried to defend the theory of Virrarert, after having made new experiments and no less a person than von brnrine doubted the exactness of the generally assumed opinion, no one will be surprised that criticism soon followed. Whilst I was working in the laboratory of Prof. SPRONCK, to whom I offer my thanks for his continual interest in this research, repeating the experiments of VANSTEENBERGHE and Grisrz, several treatises appeared on this subject. First of all AscHorr *) advanced 1) Annales de l'Institut Pasteur, 1905, p. 787. 2) Virarer: Cas rare d’anthracosis, Paris, 1862. ref. in Schmidt’s med. Jahrb. 1862, Bd. 116. 3) ArnoLp: Untersuchungen ueber Staubinhalation und Staubmetastase, Leipzig, 1885 4) Sitzungsber. der Gesellschaft zur Bef. der Ges. Naturwissenschaft, Marburg, 13 Juni, 1906, ( 674 ) the opinion that there must have been technical mistakes in the experiments of VANSTEENBERGHE and GRisez; some time afterwards he was enabled to convince himself of the incorrectness of their opinion by his own experiments '). Mironesco ?) after bringing fine particles into the stomach of rabbits, was not able to recover them in the lungs. In August 1906 VANSTEENBERGHE and SONNEVILLE*) described a new series of experiments which confirmed the results of VANSTEEN- BERGHE and GRiskz. Fine particles which were brought into the mouth with a catheter were already to be recognised in the lungs after a lapse of 5 or 6 hours. Soon afterwards the opinion of VANSTRENBERGHE and GRISEzZ was opposed by two authors: Scauze*) in a temporary publication con- eluded that the pulmonary anthracosis could not proceed from the resorbing of fine particles from the intestines and Prof. Spronck com- municated shortly afterwards at the 5% International Conference on Tuberculosis the results of some of the following experiments, which were adverse to the results, gained by VANSTEENBERGHE and GRiskzZ. In a more extensive treatise ScHurze°) demonstrated further how substances are lightly aspirated into the lungs either by administering them with the catheter or by ordinary eating. A rabbit however, had received within two months the total quantity of 200 grams of vermillion through a gastrotomy, yet no trace of vermillion was to be found in the lungs. On the other hand some investigators took the part of VANSTEEN- BERGHE and Grisrz: Petit’) brought carbon particles into the stomach of six children who were in an advanced state of tuberculosis or athrepsy and after a post-mortem examination he found pigment in the lungs in three of them and Hermann‘), on the authority of experiments, esteemed an intestinal origin of the lung-anthracosis possible, but compared with the inhalation-anthracosis of very inferior significance. Afterwards the results of VANSTEENBERGHE and Griskz were empha- 1) Braver’s Beiträge zur Klinik der Tuberculose, 1906, Bd VI, Heft 2. 2) Compt. rend. de la Soc. de Biol. 1906, T. 61, N°. 27. 3) Presse médicale, 11 Août 1906. 4) Münchener Med. Wochenschr. 1906, N° 35. 5) Zeitschrift für Tuberculose, October 1906. 6) Presse médicale, 13 Octobre 1906. 7) Bulletin de Académie royale de médecine de Belgique, Séance du 27 Octobre 1906. La Semaine médicale, 1906, NO 44. ( 675 ) tically contradicted from various sides. (Conn), REMLINGER*), Basser"), Kiss et Lossruin*), Berrzkn')). Some of the above mentioned considered the normal anthracosis in test-animals as a source of mistakes, which VANsrTEENBERGHE and Grisez had not taken into account whereas others described the aspiration also as a source, which might give rise to wrong con- clusions. Meanwhile VANSTEENBERGHE and Grisuz, supported by CaLMerrn °) maintained their opinion. They explain the negative results of their opponents in the following manner: some allowed too much time to pass between the introducing of carbon particles into the stomach and the killing of the test-animals, because after 48 hours the pigment would have almost completely disappeared from the lungs; others used rabbits or too young cavies as test-animals, in which the fine particles are almost wholly retained by the mesenteric glands. With a view to this last remark I wish to publish the following experiments, because I have taken into account the age of test- animals as well as the time which passed between the introduction of the fine particles and the killing of the animals. To me it also appeared that the physiological anthracosis is a factor which must be considered, for among all my test-animals, cavies as well as rabbits, black pigment was found in the lungs. Among some animals this spontaneous anthracosis was rather decided, with others very minute. As a rule there was much less pigment in the lungs of my rabbits than in those of the cavies. The physiological anthracosis impedes as a matter of course the experimenting with black substances. Besides carmine, vermillion and ultramarine, I have also used Indian ink and soot, because after microscopic investigation it appeared that the first mentioned matter, even after being intensively rubbed in a mortar, was not as fine as the particles of carbon of the last mentioned. In order to control the experiments of VANSTEENBERGHE and GRISEZ 1) Berliner Klin. Wochenschr. 1906, N° 44 und 45. 2) La Semaine médicale, 1906, N° 45. 3) La Semaine médicale, 1906, N’ 47. 4) Bulletin médical du 21 Novembre 1906. La Semaine médicale, 1906, N° 48. 5) Virchow’s Archiv, Bd. 187, Heft 1. 6) Compt. rend. des séances de l'Académie de Sciences, T. 143. p. 866. Compt. rend. de Ja Soc. de Biol. T. 61, p. 548. La Semaine médicale, 1906, N°. 50. ( 676 ) the test-animals were killed already 5—48 hours after administering the forementioned substances. Some cavies (experiment n°. 1—5) had eaten bread, mixed with soot, Indian ink or carmine. After the dissection of the animals, the lungs showed only the ordinary physiological anthracosis, but car- mine was to be seen neither in the lungs nor in the bronchial glands. One of these animals (experiment n°. 4) had evidently aspirated soot, for in many bronchi and corresponding alveolars, foodparticles and soot were distinctly seen in large quantities. Also after “introducing various matters with the catheter into the stomach of rabbits (experiment n°. 6—10), aspiration was observed once (experiment n°. 10), whereas among other animals only the normal pigmentation was present. . In order to prevent aspiration with certainty, tracheotomy was performed with three rabbits and after that a suspension of carmine was brought into the stomach with the catheter (experiment n°. 11 -—13); for the same purpose among some cavies I injected coloured particles into the distal part of the oesophagus which was cut through and then bound up (experiment n°. 14—18). After dissecting no traces of coloured particles were to be found neither in the lungs nor in the bronchial glands. Further with different cavies the fine particles were directly brought into the intestines after laparotomy (experiment n°. 19—35). Neither was then any of the coloured matter. to be found in the lung-tissue nor in the bronchial glands, whereas everywhere else nothing was to be seen except normal anthracosis in varying intensity. Among some experiments I noticed that coloured particles which were injected directly into the intestines, were later on to be found also in the stomach, in the oesophagus and in the pharynx, sometimes in large quantities (experiment n°. 21, 22, 29 and 30). In the phlegm of the trachea the coloured particles could be distinctly seen some- times with the use of the microscope (experiment n°. 21 and 29), whilst once (experiment n°. 29) the easily recognisable ultramarin- grains were to be seen even in the phlegm of the chief bronchi. It is quite probable that the animals in agony had aspirated these sub- stances from the pharynx, for, according to NENNIGER') e. g. bacteria too are often aspirated from the pharynx in agony. The question is now, how came the matter from the pharynx into the intestines. Was it by a motion of the fine particles in a 1) Zeitschr. f. Hygiene u. Infectionskrankheiten, Bd. 38. ( 677 ) proximal direction, as e. g. GRÜTZNER!) describes this for fine particles in the intestines and as Kast?) has also shown for the oesophagus, or, had the animals eaten their own faeces?*) — - In order to solve this question, four cavies were carefully wrapped in a bandage, after ultramarine had been brought into the intestines so that eating the faeces was quite impossible (experiment n°. 82—385). It now appeared that the ultramarine had come some way proximal from the place of injection, but in the oesophagus, in the pharynx and in the chief bronchi no ultramarine was discernible. From this I suppose that the ultramarine had simply come into the pharynx owing to the eating of faeces and not through a proximal motion of the fine particles *). From my experiments I conclude that the. pulmonary anthracosis does not originate through taking up fine particles from the intestines. It may be acceptable a priori, that fine particles can be taken up in the intestinal mucous membrane and can get into the lungs along ductus thoracicus and right heart, but this phenomenon is with regard to the pulmonary anthracosis of not so much importance, as VANSTEENBERGHE and Grisez have supposed. Evidently these investi- gators have given sufficient attention neither to the physiological anthracosis of the test-animals, nor to the aspiration of the coloured particles which cannot be quite prevented, not even, as is mentioned above, by direct injecting the matters into the intestines. If the physiological anthracosis originated by taking up carbon particles from the intestines, not only the mesenterial glands but also the marrow and the milt had to contain much carbon pigment, because firstly it cannot be understood how carbon parts should pass the mesenterial glands without leaving distinct traces of their passing behind them and on the other hand there is no possible reason why the carbon particles to a great extent should not pass through the capillaries of the lungs and deposit in the marrow and the milt. 1) Archiv. f. d. Ges. Physiol. (Pflüger). Bd. 71. 2) Berliner Klin. Wochenschr. 1906, N° 28. 3) When starving cavies and rabbits usually eat their own faeces, it also often occurs when they have sufficient food. Swirsxr: Archiv f. exper. Path. und Pharm. 1902, Bd. 48, 4) UrrENHEIMER, after injecting a suspension of prodigiosusbacilli into the rectum of rabbits, ncticed a motion of the bacilli in a proximal direction; they ascended up to the pharynx and from thence they were sometimes aspirated into the lungs. Deutsche Med. Wochenschr. 1906, N°, 46, ( 678 ) Description of the Experiments. 1. Cavy 650 grams. First 24 hours without food, then for 24 hours exclusively dough and soot, then killed. Results: Macroscop. intestines much soot, lungs grey with small black spots, especially in the upper lobes, bronchial glands distinetly pigmented. Mieroscop. In the interstitial spaces of the lung-tissue are many cells with black pigment especially under the pleura. A very small quantity of it is also found in the alveolars and in the bronchi, The bronchial glands contain a great many cells with black pigment. 2. Cavy 200 grams. For 48 hours exclusively dough and soot, then killed. Results: Macroscop. intestines much soot, lungs and bronchial glands pale; microscop. lungs and bronchial glands few cells with black pigment. 3. Cavy 760 grams. First 24 hours without food, then 5 eem. of Indian ink in dough, killed after 24 hours. Results: as in experiment 1. 4. Cavy 350 grams. First 24 hours without food, then for 48 hours exclusively dough and soot, then killed. Results: Macroscop. intestines much soot, lungs many black spots and points, bronchial glands pale; microscop. there are foodparticles mixed with soot in many bronchi and alveolars. For a part the soot has already been enclosed in cells, many cells have already penetrated into the interstitial spaces. No pigment is to be seen in the bron- chial glands (so in this experiment the coal was aspirated during life; not in agony). 5. Cavy 400 grams. First 24 hours without food, then 0,5 grams of carmine in dough ; killed after 48 hours. Results: Except in the intestines no carmine can be found. 6. Rabbit 1.75 K.G. For three days 100 mer. of soot is brought into the stomach by means of a catheter; killed after 24 hours. Results: Macroscop. lungs and bronchial glands pale; microscop. few cells with black pigment are to be seen in the interstitial spaces of the pulmonary tissue. (679) 7. Rabbit 2 K.G. For three days totally 2,9 grams of soot is brought into the stomach with the catheter; killed after 24 hours. Results: as in experiment 6. 8. Rabbit 2 K.G. A suspension of 2 grams of carmine in water is brought into the stomach with a catheter; killed after 48 hours. Results: Except in the intestines, carmine is not to be found. SP Rabbit’2:75, KG. 50 grams of charcoalpowder, suspended in water, is brought into the stomach with the catheter; killed after 24 hours. KJ . Results: as in experiment 6 10. Rabbit 3 K.G. A suspension of 40 grams of charcoalpowder is brought into the stomach with the catheter; killed after 24 hours. Results: Macroscop. lungs show black spots especially after dissect- ing them; the bronchial glands are faintly pigmented; microscop. fine carbon particles and also coarser carbon pieces can be seen in many alveolars. Carbon can be shown neither in the larger bronchi, nor in the trachea; the bronchial glands show some pigment-cells. The presence of the coarser carbon parts in the alveolars made the diagnose ‘‘aspiration” very easy. 11. Rabbit 5 K.G. After tracheotomy 9 grams of carmine is brought into the stomach with the catheter. About 18 hours afterwards the animal chokes, as phlegm has gathered in the canule. 12. Rabbit 4.25 K.G. After wacheotomy 8 grams of carmine is brought into the stomach with the catheter; the animal is killed after 24 hours. 13. Rabbit 3.5 K.G. After tracheotomy 8 grams of carmine is brought into the stomach with the catheter. Killed after 48 hours. Results of the experiments 11, 12 and 13: Except in the intestines I could find nowhere carmine in the body at the microscopical inves- tigation; in the lungs and bronchial glands black pigment is present. 14. Cavy 400 grams. The oesophagus was freeprepared and cut through. Through the lower part 5 gram of vermillion was brought into the stomach. Then the lower part of the oesophagus was bound up whereas the 46 Proceedings Reyal Acad. Amsterdam. Vol. IX. ( 680 ) upper part was fastened with its opening in the wound of the skin. Killed after 5 hours. 15. Cavy 720 grams. 10 grams of vermillion were injected as in experiment 14. Killed after 6 hours. 16. Cavy 720 grams. 7 eem. of a suspension of vermillion in gum arabic was injected as in experiment 14. Killed after 5 hours. 17. Cavy 400 grams. 2 grams of carmine were injected as in experiment 14. Killed after 5 hours. 18. Cavy 860 grams. + grams of carmine were injected as in experiment 14; killed after 6 hours. 19. Cavy 790 grams. After laparotomy 4 grams of vermillion (in suspension) were brought into a twist of the intestines; killed after 18 hours. 20. Cavy 620 grams. 6 eem. suspension of vermillion in gum arabic was brought into the small intestin as in experiment 19; killed after 19 hours. 21. Cavy 750 grams. 10 eem. suspension of vermillion in gum arabic was brought into the small intestin as in exp. 19; killed with chloroform after 18 hours. 22. Cavy 610 grams. 5 ecm. suspension of vermillion in gum arabic was brought into the colon as in exp. 19; killed after 18 hours, by abruptly decapi- tating in order to prevent vomiting in agony. Results of the experiments 14—22: In the lung-tissue and in the bronchial glands no vermillion resp. carmine was to be found. At experiment 21 the vermillion could also be shown in the stomach, in the oesophagus and in the pharynx while some grains could be shown in the phlegm of the trachea. At experiment 22 vermillion could also be found in the stomach, oesophagus and pharynx whereas in the trachea no vermillion was to be seen. (At the experiment 19 and 20 stomach, pharynx ete. were not investigated). 23. Cavy 700 grams. After laparotomy 5 cem. of Indian ink is brought into the small intestines. Killed after 18 hours. Results: as in experiment 1. ( 681 ) 24. Cavy 880 grams. After laparotomy 5em. of Indian ink is brought into the coecum. Killed after 18 hours. Results as in experiment 1 (the pigmentation is somewhat less intensive). 25. Cavy 750 grams. After laparotomy 5cem. of Indian ink is brought into the small intestines. Killed after 18 hours. . Kesults : macroscop. Lungs and bronchial glands pale microscop. few pigmentcells. 26. Cavy 700 grams. After laparotomy Seem. of Indian ink is brought into the coecum. Killed after 18 hours. Results as in experiment 1. (here the pigmentation is more intensive.) 27. Cavy 730 grams. After laparotomy Seem. of Indian ink is brought into the small intestine. Killed after 18 hours. Results as in experiment 25; one of the mesenteric glands con- tains carbon parts which are also to be seen microscopically. 28. Cavy 750 grams. After laparotomy 5 cem. of Indian ink is brought into the colon at 20 em. distance of the anus. Killed after 18 hours. Results as in experiment 1. 29. Cavy 650 grams. After laparotomy 4 eem. of a suspension of ultramarine in 0.9 °/, NaCl is brought into the small intestines. Killed with chloroform after 18 hours. Results: The ultramarine is in the intestines, in the stomach, in the oesophagus and in the pharynx, while some grains can be traced in the phlegm of the trachea, and in that of the chief broncii. The pulmonary tissue and the bronchial glands are free of ultra- marine. 30. Cavy 850 grams. After laparotomy 4 ccm. of a suspension of ultramarine in 0.9 °/, NaCl is brought into the small intestine. Killed after 17 hours with chloroform. Results: as in experiment 29; in the phlegm of the trachea and in that of the bronchi however no ultramarine was to be found. ( 682 ) 31. Cavy 820 grams. 4 eem. of ultramarine is administered as in experiment 30. Killed after 16°/, hours. Results: No ultramarine can be found, except in the intestines. 32. Cavy 360 grams. 4+ eem. of ultramarine is brought into the intestines as in expe- riment 30. After this the animal is carefully wrapped up so that it can get no faeces into its mouth and cannot lick itself. After 6 hours the animal is decapitated abruptly in order to pre- vent vomiting in agony. Results: the ultramarine is in the small and in the large intestines, also somewhat proximal from the place of injection. In the stomach, in the oesophagus, in the pharynx and in the phlegm of the chief bronchi no ultramarine can be traced, 33. Cavy 750 grams. Treated as in experiment 32. Killed after 16 hours. Results: as in experiment 32; here some grains of ultramarine are in the stomach. 34. Cavy 475 grams. Treated as in experiment 32. Killed after 12 hours. results: as in experiment 32. 35. Cavy 540 grams. Treated as in experiment 32. Killed after 6 hours. Results: as in experiment 32. (March 28, 1907). KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM. PROCEEDINGS OF THE MEETING of Saturday March 30, 1907. DOG (Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige Afdeelmg van Zaterdag 30 Maart 1907, Dl. XV). GON TEELEN 2s: J. P. vAN DER Srox: “The treatment of wind-observations’’, p. 684. F. M. Jancrer: “On the anisotropous liquid phases of the butyric ester of dihydrocholesterol, and on the question as to the necessary presence of an ethylene double bond for the occurrence of these phenomena”. (Communicated by Prof. A. P. N. FRANCHIMONT), p. 701. P. van RomBuren and A. D. MAURENBRECHER: “On the action of bases, ammonia and amines on s. trinitrophenyl-methylnitramine’’, p. 704. W. H. Junius: “Wave-lengths of formerly observed emission and absorption bands in the infra-red spectrum”, p. 706. ©. H. Wixp: “A hypothesis relating to the origin of Röntgen-rays”, p. 714. J. H. Meersure: “On the motion of a metal wire through a piece of ice”. (Communicated by Prof. H. A. Lorentz). p. 718. J. D. van DER Waats: “Contribution to the theory of binary mixtures”, II, p. 727. J. D. van DER Waars: “The shape of the empiric isotherm for the condensation of a binary mixture”, p. 750. H. KAMERLINGH ONNES and C. Braak: “Isotherms of diatomic gases and their binary mixtures. VI. Isotherms of hydrogen between — 104° C. and —217° C.”, p. 754, (With 2 plates). H. KAMERLINGH Onnes and C. Braak: “On the measurement of very low temperatures, XIV. Reduction of the readings of the hydrogen thermometer of constant volume to the absolute scale”, p. 775. (With one plate). H. KAMERLINGH Onnxes and W. H. Kerersom : “Contributions to the knowledge of the 4-surface of vAN DER Waars. XV. The case that one component is a gas without cohesion with mole— cules that have extension. Limited miscibility of two gases”, p. 786. (With 2 plates). 47 Proceedings Royal Acad. Amsterdam. Vol. IX, ( 684 ) Meteorology. — “The treatment of wind-observations.” By Dr. J. P. VAN DER STOK. (Communicated in the meeting of February 23, 1907). 1. When working out wind-observations we directly meet with the difficulty that a method holding generally, in which the charac- teristics of a wind distribution come to the fore in condensed form, does not exist. The discussion held for many a year concerning the desirability or not of an application of Lampert’s formula, i. e. of the calculation of the vectorial mean of velocity or force has not led to a definite result and the consequence is that for regions where trade- and monsoon winds prevail the calculation of this mean can be applied, not for higher latitudes, so that here we have to judge by extensive tables of frequencies of direction and mean velocities, independent of direction. When working out the wind-observations made at Batavia I did not hesitate to make an extensive use of this formula; the same method has been followed in the atlas for the Kast Indian Archipelago; but in order to give at least a notion of the value of the velocities annulling each other here I have added to the resulting movement (called by Hann windpath) a so-called factor of stability. If namely the wind were perfectly stable, the vectorial mean would be equal to the mean independent of the direction and the stability would amount to 100 °/,, which percentage becomes smaller and smaller according to the direction of the wind becoming more variable. So here attention is drawn to the fact, that a part of the observations is eliminated, but it is not indicated what character this vanishing part has which becomes chief in our regions. In the climatological atlas lately published of British India the same method is followed; in the “Klima Tabeller for Norge’ Moun gives but the above mentioned tables without calculation of the vectorial mean, which is, indeed, of slight importance for this climate. The same uncertainty is found in the graphical representation of a wind distribution by so-called windroses; almost everyone who has been occupied in arranging books of prints has projected wind- roses of his own; some of those roses, as e.g. in the “Vierteljahrs- karte für die Nordsee und Ostsee” published by the ‘Deutsche Seewarte’, show only the frequencies of direction without velocities ; in others, as e.g. those shown in the above atlas of the East Indies, each direction is taken into account with the velocity belonging to it as weight, so that mean velocities are represented. All these roses ( 685 ) furnish discontinuous quantities and. change their aspect according to their boundaries being taken differently. In Bucnan’s general meteorological atlas no roses are projected, only arrows indicating the most frequent direction without heeding the force, and in the “Segelhandbuch für den Atlantischen Ozean” published by the “Deutsche Seewarte” for higher latitudes where the wind is variable the use of wind-observations is entirely done away with and arrows have been drawn in accordance with the course of the mean isobars on account of the law of Buys Barror, where a constant angle of 68° between gradient and direction of wind has been assumed. This short survey of the manner in which in the most recent ‘standard works this problem has been treated may show that indeed there is as yet no question about a satisfactory solution, as has already been observed. The aim of this communication is to hit upon a general method of operation and representation of an arbitrary wind distribution in which to the variable part also justice is done, whilst the gra- phical representation has a continuous course and shows at a glance the five characteristic quantities which mark each wind distribution and which may be, therefore, called the wind-constants. The method proposed here is founded on the basis of the calculus of probability, but it is important to notice that it is not at all bound to it; at the bottom it is the same which is generally applied in the treatment of directed quantities: distribution of masses and forces in mechanics, the theory of elasticity, the law of radiation and the theory of errors in a plane. 2. A wind-observation can be represented by a point in a plane such that the distance to an assumed origin is a measure for the velocity of the wind (or force) and that the angle made by the radius vector with the Y (North) axis counted from N. to E. indicates the direction. If in this way all observations, V in number, are drawn and if we think that to each point an equal mass is connected, then in general the centre of gravity will not coincide with the origin selected; its situation may be determined by the quantities R, and ea. The distribution of the masses around the centre of gravity, is then characterized by the lengths J/ and M’ of the two principal axes of inertia and the angle 3 enclosed by the axes M and Y. As is known the five constants by which such a system is charac- terized can be calculated according to this purely mechanic notion by determining the moments M, and J/, with respect to the axes 4+7* ( 686 ) and the moments of inertia M,* and M,? and M,,, which furnish the five equations necessary for the calculation of the unknown quantities. We arrive at quite the same equations when the distribution of the winds according to direction and velocity is regarded as a system of accidental, directed quantities in a plane. The centre of gravity then represents according to size and direction the constant part of the wind which is supposed to be connected with all observations and of which, therefore, the probability is equal to unity ; the axes of inertia become principal axes of probability and the lengths M and M’ are replaced by the reciprocal lengths Ah and h’, so that tat eect — IM? ’ = 2M" . . . . . . ( ) The sum of the masses is put equal to unity and for the proba- bility that an observation lies between the limits R and R+ dk of velocity and 6 and 4 + d as far as direction is concerned the expression holds es ER RaRGO,. con elt bg where: fR‚O)=h" [Reos (O—B)—R, cos(a—B)]’ +h"? [| Rsin(6 —B)—R,sin(a—8)]*.(3) In the language of the theory of errors \f,,@) would be the so-called constant error, M and M' the greatest and smallest projections of the mean errors. As observations of wind agree still less than other meteorological quantities with the opinion held in the theory of errors, where the constant part is regarded as the end of the operation and the variable quantities as deviations, it is desirable when applying the caleulus of probability to quantities of this kind to be entirely free of the terminology used in the theory of errors, but which would be here without meaning and which would give rise to misunderstanding. The treatment must also differ somewhat from that of erroneous quantities, it being if not impossible at least impractical to correct all the observations for the constant part. 3. As examples of treatment two series of observations have been selected from the treated material. a. Observations of wind performed at Bergen (Norway) during 20 years, 1885—1904, three times daily at 8 A.M, 2 P.M. and 8 P.M. The velocity (or force) of the wind is expressed in the so-called ( 687 ) half scale of Bravrorr \1—6) (Jahrbuch des Norwegischen Meteorol. Instituts, Christiania). 6. Observations of wind performed at Falmouth (Channel) during 17 years, 1874—1886 and 1900—-1903; the observations made in the years 1887—1899 are published in such a way as to be useless for this investigation. Observations have been used, made daily six times: at noon, + P.M., 8 P.M., midnight, 4 A.M. and 8 A.M.; the velocity of wind is expressed in English (statute) miles an hour (Hourly readings obtained from the selfrecording instruments etc. London). With respect to the force of the wind estimated at Bergen is to be noticed that in this communication these scale-values are regarded not as forces but as velocities, although in reality they are neither one nor the other. According to a recent extensive investigation *) the ratio of the Beaufort values to corresponding velocities can be indicated by the following numbers Breavurort velocity ratio Beaurort velocity ratio meters a second meters a second 0 1.34 — 6 10.95 1.83 1 2.24 2.24 d 13.41 1.92 2 3.58 1.79 8 16.09 2.01 3 4.92 1.64 9 19.67 2.19 4 6.71 1.68 10 23.69 2.34 5 8.72 1.74 As the various velocities do not appear in an equal number the total mean out of these ratios would not give a fit factor of reduction for mean Braurort-values; so a certain weight must be assigned to each separate ratio. For this the frequencies have been used of the 36000 wind-velocities observed at Falmouth calculated for a whole year; in this way has been found for the reduction-factor 1.83; the English measure, miles an hour, can be reduced to m.a.s. and Bravurort scale-values by means of multiplication respectively by 0.447 and 0.244. 1) The Beavrort scale of windforce. Report of the Director of the Meteor. Office upon an Inquiry into the Relation between the Estimates of Windforce according to Admiral Braurorr’s Scale and the velocities recorded by Anemometers. London, 1906. ( 688 ) 4. The calculation of the five charateristie constants of a wind distribution amounts in one respect to the integration of (2), in another respect to the means applied in this integration to a given set of observations. The integration of (2) takes place by the introduction of rectangular coordinates : 2 == Risin Over Wed, where the element RdRd@ is replaced by the element dxdy, whilst the limits which were o and O for &, 2” and O for 6, now become oo and — oo. Then the expression (2) under the sign of the integral is multiplied successively by * and zy. fy ye, y If we then put: R, cos (a—fB) =a, z= 2' sinB + y' cos B, R, sin (a— B) = d, y = «x' cos B — y' sin B, the variables 2 and y' can be separated and the integration can be done; in this way we find for the determination of the five quantities to be obtained the five equations : M, =acosB —bsnB, M, =asinB + beos B cos° B sin? B M‚ = on Dh + a? cos? B + b? sin? B — ab sin 2 ‚ns 2 My TE SEE psp Worden (> © 2M., = En — an) sin 28 + (a? — b?) sin 2B + abcos 28 out of which, on account of (1) M, =R,cosa, M,=R, sin a \ Mt + My? — [M) + HT = U + M" ee. M;? — Mf? — (LY UTM ears, 2M,, — 2M,M, = (M* — M'*) sin 2B ( 689 ) TABLE I. Frequencies of the wind. Bergen. June. In half Beavrorr scale-values. 0 | 1 2 3 4 5) Sum € 261 — — —| — — 261 N 59 30 29 5 2 125 NNE 6 6 1 — — 13 NE 6 2 =| — Ee 8 ENE 3 2 oe = 5 E iS she eth 13 ESE 5 4 4 =e) NE Ls 5 SE 24 3 1 | — — 28 SSE 40 16 3 | — =~ 59 5 115 54 22 6 — 197 SSW 56 39 15 4 — 114 SW 25 10 2 2 — 39 WSW 9 2 —-| — -- 41 W 98 26 5 — — 129 WNW 99 24 1 — — 124 NW 190 51 6 | — — 247 NNW 246 | 118 6 | 12 — 422 Sum 961 | 993 | 384] 131 | 29 . 2 | 1800 5. In order to apply the formulae (4) to a given set of obser- vations we must compose for each period, e.g. each month, in the first place a table of frequency of direction and velocity, which can be easily done. In Table I such a composition has been given as an example. Further out of this table have been calculated the products of these frequencies f with the scale-values R, the latter counted double, so that the products have been expressed in the ordinary Bravrort scale; finally these products have been once again multiplied by the corresponding scale-values (/R?); in this simple way we find the sums. ( 690 ) TABLE II. 620 1336 4560 The sums fR, multiplied respectively by cos@ and sin@ and divided by 1800, immediately furnish the quantities M, and M, ; the sums /R? must be multiplied successively by cos? 0, sin? @ and sin @ cos 6. It is easier to multiply the latter sums by cos 26 and sin 26; if the total mean is S, we find: M,? = fR? cos? A= 34 S + 3 fR* cos 20 My? = JR? sin? 6 = 5 S — 4 fR? cos 20 2 May = FR? sin 26. So the whole operation greatly resembles the calculation of Fourier terms; indeed, also by the way of operation indicated here an analysis of the movement of the air is obtained. In the Tables III and IV we find the values of the wind-constants calculated in this way; besides the five characteristic quantities we find still given as quantities practically serviceable for various ends: V M?—M" rt M M’ represent the half principal axes, (Lt, and a’) the resultants of the squares of the velocities giving an image of the mean flux of energy, V the mean velocity independent of the direction, V? the mean square of the velocity independent of the direction, i.e. a measure for the total energy; this quantity is according to (4) analogous to the square of the mean error, not corrected for the constant part, in the theory of errors, N the number of used observations. e , the excentricity of the ellipse of which J/ and January February March April May June July August September October November December Year ( 691 ) TABLE IIa Constants of the wind. Bergen 1885 1904. In BEAUFORT scale-values. oe 1.84 479° 4.51 180 1.16 183 0.40 25 0.68 284 de OF 302 0.93 276 0.73 251 0 97 212 1.10 182 1.51 179 Leite 179 0.85 203 M ow wb cc = 3 14 2.97 M' fe ial ol AN TABLE IIIb Constants of the wind. Bergen 1885—1904. In BEAUFORT scale-values. January February March April May June July August September October November December Year 1860 1800 1860 21915 ( 692 ) TABLE IVa. Constants of the wind. Falmouth. 1874—1886, 1900—1903. In Eng. miles an hour. January 15.20 13.93 73° 0.400 February 14.08 13.25 164 0.339 March 15.02 13.26 67 0.470 April 13.70 12.24 72 0.454 May 12 02 41:52 40 0.286 June 11.19 10.07 |, 158 | 0.436 July 10.39 8.92 155 0.507 August 10.48 916 | 82 0.363 September 11.05 | 10.67 | 164 0.260 October 13.51 13.03 | 81 0.266 November 13.75 13,03: | 3 0.318 December 13.69 12.98 22 0.318 Year 12.60 12.43 | 96 0.166 | TABLE IVb. Constants of the wind. Falmouth. January 2821 February 159.9 203 2675 March 88.8 241 2930 April 23.0 178 2879 May 54.4 OM 3110 June 92.4 252 3015 July 422.2 252 3060 August 122.0 249 3154 September 83.7 224 3047 October 78.5 223 3154 November 114.2 237 3053 December 148.6 233 2888 Year 98.2 229 35816 ( 693 ) A closer discussion of the results arrived at in this way may for shortness’ sake be left out; however, the observation is not super- fluous that the two examples represent two types, a reason why they were chosen. At Bergen the ellipse of the variable winds is very constant of shape and the excentricity is very great; at Falmouth the difference between MM and M’ is always very slight and the differences found there are evidently to be regarded rather as accidental arithmetical results than as facts, the angle 3 being subject to great and irregular oscillations; evidently the ellipse approaches a circle, so that in form (2) we may put h=/'. This leading to a considerable simplification of the formula, these observations at Falmouth are eminently fit for comparison of the results of calculation and obser- vation, whilst also the fact that here real velocities have been observed with well-verified instruments, makes this series very favourable. 6. The expression (2) shows: the-probability that an observation lies between the limits R and R+ dk, 6 and@é+dé6; the same expression without the element RdRd6 indicates: the specific proba- bility of a wind (R,6) ie. the probability with respect to the unity of surface when one imagines this surface to be small. If we put for simplification : 2th? = 2p, h? —h? =2q, R,?(p — q cos 2 (a — B)) =u (p — qcos 2 (8 —B)) =v, s*= R,*(p? + g? — 2pq cos 2 (a —8)) sin a sin (a — 2 scos (0 — p)=À tang p HE EEN , p cos a — q cos (a — 2 B) then (2) takes the form : it: @ EN b RARA san a eee ae Jt If here we put: Rip = 2 TE en et be ND then it follows out of the above formulated definition that the specific probability of all observations lying on the circumference of the excentric ellipse (6) is the same and equal to: Kn? tlr. Jt The probability that the velocity of the wind does not surpass the value f, expressed by (6) in function of 6, in other words the number of observations which are to lie within the area of the ellipse, is ( 694 ) found by integrating (5), first with respect to R between the limits R, and O, then with respect to 6 between 27 and 0. For the, simple case A, = 0, so also. » =O and 2=0; the first integration gives immediately Vp?—q? Iet ld 2v and as Qn oe (ee Dat NED RE 0 the probability to be found becomes simply : PSOE Ue tech. | td eae ee and the number of observations lying inside the circumference of the ellipse (6) : N (le ). This amount remaining the same whether we regard the ellipse (6) from the excentrie origin or from the centre, ie. for R, = 0, if with the integration the limits are changed correspondingly, the expression (7) must also be accurate when &, is not equal to zero and must thus hold in general. Indeed, an other simplification, namely q = 0 (which is applicable to the results for Falmouth) leads to a set of definite integrals, which can be evaluated and which confirm this conclusion. Amongst the series of ellipses represented by (6) two are remarkable; if we assign to c the value 0.5, then on account of (1) the half axes of the ellipse become equal to the greatest and smallest projections J/ and M’ of the mean velocities, so that the ellipse (6) then represents what we might call the specific or typical windellipse, thus a kind of windrose, in which the characteristic qualities of the wind-distribution under consideration inmediately become conspicuous. The radius vector Zi, drawn to an arbitrary point in the circum- ference is given in the direction determined by that choice by the equation : 0 UR iv sR, 2-2 on He. The probability that a velocity does not surpass this value is: 1 — ea = 0.389347. ( 695 ) So among a thousand observations there will be 393 lying inside this typical ellipse whilst the specific probability of each of the velocities Fm is: 0.6065 VPL JT In the given diagram such a typical windellipse is represented for Bergen in the month of June by the dotted line. the vector OC represents here the constant part (R,, a), the half axes are equal to M and MM’, and the angle NOM=8; one millimeter corresponds to */,, Braurort scale- value or to */,, < 1.83 = 0.275 meter a second. If necessary this diagram might be am- plified with two circles, one of a radius VM + mM", representing tle mean monthly wind velocity corrected for the constant part, the other described with radius VM 4 MT FAM) + U)’, which is according to (4) a measure for the mean total velocity, corresponding to the square root of the quantity V? of the tables III and IV. An other remarkable ellipse which might be called the probable windellipse is obtained by requiring half of the observations to lie within its dominion; we have then to determine c in such a way that hte ss 7). 83 te = Oa so that the axes of this ellipse are V2e = 2 X 0.8326 = 1.177 times longer than those of the typical windellipse ; the number 0.8326 is a quantity known in the theory of errors in the plane. 7. The frequency of the windvelocities, setting aside the direction, cannot be represented in a finite form; we can arrive at a form serviceable for comparison with the observation by writing (5) thus: feeb PR Bp 2B RdRdô, . . . (8) JT ( 696 ) by developing the last exponential factor and then by expressing the powers and products of cosines in cosines of multiples. It is clear that when integrating (8) with respect to 9 from 22 to 0 only those terms are left which are independent of @ and which appear with the common factor 27. The expression to be found for the probability that a velocity lies between the limits R and R-+ dR then becomes: 2 Vp gerela Rha JRER .” 1) where: "Wescott bah a, =4,/2" + gs?/2! cos 2 (p—B) + 2*/(2!), a, = 9787/2? + ge*/3! cos 2 (~—B) + 8°/(8/)’. For Falmouth, where as was noticed above q can be put equal to nought these coefficients become simply : g2n ni 2 an — 5 ; ml s=pR, , p=pR, VvE Pe, 4= pR, cos (O-—a). (10) and farther In practice it will frequently be only necessary to calculate a few of these coefficients; if we put: q/p = &, the integration of (9) between the limits m and 0 leads to the expression : Vises \ Rn een 2la, . dla, (levy. (1+ 24-24 —F4+.... P 2 Pp pmte pm G is 2/a, it ) Ae JE (11) if A mene ee ai pmte? (2!a, Sn En DL NER RN En AOR: As for m == this expression must become equal to unity, we have: fe a, ii 2!a, ie el: sn na or, for the case q = 0, (11) becomes : ( 697 ) 1—e— pn? pm'en = = aa ‘ia he ek Ngee DRA aT) 2m e—pn avs (: — ET — “1 —) ete. 2! p from which is immediately evident that in many c REET three first terms are sufficient, so that then the calculation of the coefficients can be entirely avoided, or at most only a, must be taken into account; for generally u is small, so that already 1—e—? will be a small quantity. If g is not small the calculation becomes rather tedious. 8. To find expressions for the quantities V and V*, the mean velocity and the mean square of the velocity independent of the sign, we have to multiply (9) successively by R and PR? and to integrate between the limits oo and O which, with the well known fundamental equation, leads to the expressions : 1=J/4 1 = eh A ( + opt Gp ter + aie 21 ees sy Malas Toa V —=— en Val 13 Gat (2p)? aa …) ze 103) 2a, 24a, 2.4.64, ) \ vit! oo at ) Se (2p)? (GaP ae A =V1—8 et. / 9. For the calculation of the frequency of the directions independent of the velocity we have first to integrate (5) with respect to R between the limits « and O and then with respect to between the desired limits 6; the mean velocity as function of the direction is found by the application of the same operation to (5) after multiplication by R. It is then easy to give to a frequency-formula found in this way the form of a Fourtr series. For brevity we treat here only the case that ¢=O and the angle-limits are a to 0. By putting ( 698 ) we get (5) reduced to the form: VeaT ab oel (le aa ee NE) If g=0, so that the formulae (10) hold good, we then find for the desired frequencies in the two easterly quadrants RW p sine 1 Ef b+ fe ge AA Jt From this formula it is evident in what way and in what degree the asymmetry of the distribution is dependent of A,, @ and p. 10. The application of the given criteria has been made for Falmouth and the four seasons: Winter: December, January, February, number 8384, p = 0.00258, q = 0.00004 Re 3.22, a == 222°8' Spring: March, April, May, number 8949, = 0.00298, q = 0.00028 = eel, a = 250°25' Summer; June, July, August, number 9229, p = 0.00485, g = 0.00029 Ri 5.68, a = 251°22!' Autumn: September, October, November, number 9254, p = 9.00313, q = 0.00004 i = 3.80, a = 239°16' For each series the number of observations is reduced to 10.000 and everywhere we have put g=0, the calculated values are accordingly accurate as far as the fourth decimal. In Table V we have compared the observed frequencies of wind- velocities independent of direction with those calculated according to formula (12), from which it is evident that the differences havea clearly systematic course. Just as is the case with all series of errors the number of the observed small velocities is larger than would agree with the normal distribution. The differences together amount in summer to about 10°/,, in winter to 15°/,. ( 699 ) In the calculation of the frequencies of the directions independent of the velocity, the observations regarded as calms — and to these are reckoned in the English records all velocities less than 4 miles an heur — have been distributed proportionally to the frequencies of direction; furtheron the frequencies North and South are assigned for one half to the eastern and western quadrants. As is evident from the following table also in this comparison systematic differences appear; in all seasons the observed frequencies in the western quadrant are greater than the calculated ones, so that an increase of the constant part ZR, to which this uneven distribution can be attributed, would improve the correspondence. TABLE VI. Frequencies of winddirections at Falmouth for 10.000 observations. Observed |Calculated| Difference Proceedings Royal Acad. Amsterdam. Vol. LX. ( 700 ) TABLE V. Frequencies of windvelocities at Falmouth. For 10.000 observations. WINTER SBR ING SUMMER AUTUMN Miles an hour | | Observ. |Calculat.! Difference] Observ. |Calculat.| Difference | Observ. |Calculat.| Difference | Observ. |Calculat.| Difference QO — 45 45— 95 Q5—1 45 145—|95 495—945 945205 295345 345—395 395— 445 445—495 495—545 545—595 595— 760 1871 1853 1701 1466 967 680 369 199 94 417 1482 2026 2030 1650 4425 656 331 144 + 283 | — oo = Ed dt + 756 2073 569 1759 1558 039 473 201 12 + 187 + 314 — 159 ho = ez “_ | ne ++4+4+ + 945 2610 2538 1868 1164 810 936 2126 ++ +44 v ( 701 ) Chemistry. — “On the anisotropous liquid phases of the butyric ester of dihydrocholesterol, and on the question as to the necessary presence of an ethylene double bond for the occur- rence of these phenomena”. By Dr. F. M. Jancrr. (Communi- cated by Prof. A. P. N. FRANCHIMONT). (Communicated in the meeting of February 23, 1907). § 1. In order to explain the behaviour of substances which are wont to exhibit double-refracting liquid phases, some investigators have started the hypothesis that, in this kind of organic substances, it might be a question of systems formed of two components, and of equilibrium phenomena between tautomeric and isomeric modifica- tions, which would be converted into each other with finite velocity. Although it is difficult to understand how such a supposition, which is easy to propound, but very difficult to prove, could explain the numerous well ascertained facts of the regu/ar optical anisotropism of these phases, it might explain, however, at least to some extent, the peculiar irreversible transitions of phases, which I found more particularly with the esters of cholesterol and a-phytosterol, and also the hindrance phenomena noticed on that occasion’). Such a supposition, however, is perhaps of some importance for the interpretation of the brilliant colour phenomena which accompany the phase-transitions in the chelesterol esters. For a mixture, or an emulsion of substances, whose indices of refraction differ very little, but whose dispersions differ much, might, like CurisTHIANSEN’s mono- chromes, cause a similar display of colours. § 2. There is more than one cause for tautomerism (or isomerism) in the case of these cholesterol esters, for all the esters, as well as cholesterol itself, possess an asymmetric carbon atom, and in solution they all polarise to the left. Consequently, a racemisation during the esterification is by no means excluded, and we might, therefore, have a mixture of the optical antipodes. Cholesterol, moreover, possesses an ethylene double bond, so that we may also expect an isomerism in the sense of fumaric and maleic acids. § 3. As many other compounds (in fact most organic substances which are wont to exhibit these phenomena of doubly refracting 1) F. M. Jarcer, These Proc. 1906 p. 472 and 483 (29 December). 45* ( 702 ) liquid phases) possess such ethylene double bonds, one might indeed imagine that the presence thereof in the molecule is of great importance for the occurrence of the said phenomena, if not the conditio sine qua non, as the structure of the azoxy-compounds is not yet firmly established and because it may be assumed that they contain, perhaps, similar double bonds between MN and 0. Moreover, the cholesterol esters all contain three liquid phases, so that this peculiar complication might perhaps also be connected with the possibility of very intricated isomerism-phenomena of those substances. | $ 4. In order to answer these questions, I asked Prof. Dr. C. NeuBerG of Berlin to furnish me with a specimen of his synthetic Dihydro- cholesterol, to which request this savant most willingly acceded. I wish to thank Prof. NeruBerG once more for his kindness. In this Dihydrocholesterol the ethylene double bond has disappeared owing to the addition of two atoms of hydrogen, and the malenoid and fumaroid isomerism is therefore, a priori excluded. $ 5. I have prepared from this alcohol the acetic and the normal butyric esters, by means of the pure acid-anhydrides, and have examined the same as to their phase transitions. The acetic ester will be described elsewhere later on; here the butyric ester only will be discussed. As a highly important result [ may mention that the colour pheno- mena on melting and the occurrence of three liquid modifications in the normal butyrate remain unaltered as before, but that the irre- versibility of the phase-transitions is shown in a manner just the reverse as in the case of most of the cholesterol esters, e.g. the laurate. Whereas of the two doubly-refracting liquid phases of the last named substance, one is always passed over on cooling, whilst both are found on melting the solid substance, this is just the reverse in the case of the dihydrocholesterol-n-butyrate. 6. The solid phase S consists of an aggregate of very thin, colourless, and clear transparent laminae in which the plane of polari- sation makes an obtuse angle with the sides of demarcation and exhibit in convergent polarised light a hyperbole with very strong colour dispersion @ > v. B On beating, this phase S passes into a doubly-refracting liquid 5, consisting of very small, feebly doubly-refracting individuals, which in turn passes at a higher temperature into the isotropous fusion Z. ( 703 ) Of colour phenomena during one of these transitions, absolutely nothing is noticed. If, however, we start from the phase 1 and allow the same to cool, we first notice the doubly-refracting phase B, which on further cooling, amid violent sudden currents of the mass, passes into a much more strongly doubly-refracting liquid A, which on continued cooling crystallises suddenly, also amid very violent currents, to an aggregate of flat needles, glittering in vivid interference colours. These in turn, rapidly assume a spherolite structure so that the solid phase S itself appears to be also dimorphous and monotropous, as the flat needles are not reobtained on warming the spherolitie mass. The transformation of A into these needles, during cooling, is accompanied with the most vivid display of colours. Under the microscope these may be recognised by the dark-green colour of the background of the field of vision; with the naked eye, however, with incident light, that colour-display commences with a brilliant violet gradually turning into blue and finally into a radiating green when the mass crystallises. | have never noticed red or yellow colours with incident light. These phenomena return in the same order when the experiment is repeated. That the phase A really exhibits the behaviour of a stable phase p Fig. 1. Schematic p-¢-diagram for Dihydro-cholesterol-n-butyrate. ( 704 ) is also shown by the fact that, the colour having become blue or green on cooling, turns again violet on warming, so long as the solid phase S has not yet been attained. The phase is, therefore, realisable at a change of temperature in two directions. § 7. As I had but very little of the substance at my disposal, the thermometrie determinations could only be studied in capillary tubes with the aid of a magnifying glass. At 82.°1 the phase S melts to a doubly-refracting phase B which becomes clear at 86.°4 and passes into £. On cooling this isotropous fusion, it first passes properly into 5 at 86°.4, but at 84° into the more strongly doubly-refracting phase A, which may be undercooled many tens of degrees, and with retention of its violet colour, before passing into the solid phase S. Want of material prevented my determining the true solidifying point of S by inoculation; I estimate it at about 80°. Thus the positive proof has been given that the remarkable colour phenomena accompanying the melting the cholesterol esters cannot be attributed to the presence of an ethylene double bound; also that an eventual presence of fumaroid and maleinoid isomers cannot: be considered as the cause of the occurrence of the three liquids. Zaandam, 15 Febr. 1907. Chemistry. — “On the action of bases, ammonia and amines on s. trinitrophenyl-methybutramine.” By Prof. P. vaN RomBurGH and Dr. A. D. MAURENBRECHER. (Communicated in the meeting of February 23, 1907). s.-Trinitrophenyl-methylnitramine, as has been known for a long time, is decomposed at the ordinary temperature by ammonia in alcoholic solution, or on warming, by an aqueous solution of potas- sium hydroxide, or carbonate, in the first ease with formation of picramide, in the second (with evolution of monomethylamine) of pierie acid. One of us who formerly studied the reaction with bases concluded, from the occurrence of the amine and the formation of nitric acid which was also observed, that the methyInitramine which might be expected according to the equation : ( 705 ) C.H, (NO), .N-CH, +-KOH = C,H, (NO,),. OK + HN CH, | N 0, NO, might have become decomposed '). From the reaction of methylamine on tetranitropheny|l-methyInitra- mine and on trinitromethylamidomethylnitramidobenzene he after- wards concluded *) that, probably, there had been formed methyl- nitramine, meanwhile discovered by FRANCHIMONT and KroBBie *). The amount of amine formed by the decomposition of trinitro- phenyl-methylnitramine by alkalis is considerably smaller than might be expected from theory; the possibility, therefore, exists that the reaction proceeds indeed mainly in the above indicated sense. We have, therefore, taken up the problem again in the hope that by suitable modifications in the reaction, we might get at a process for the preparation of methylnitramine which would have the advan- tage of yielding this costly substance from a cheap, easily accessible material. We were not disappointed in our expectations. If trinitrophenyl-methylnitramine, which is the final product of the nitration of dimethylaniline and melts at 127°, is boiled with a 10°/, solution of potassium carbonate a brownish-red solution is obtained, which on cooling gives an abundant deposit of potassium picrate. If after filtration the liquid is acidified with sulphuric acid and again filtered off from the picrie acid precipitated and then agitated with ether, the latter yields on evaporation crystals, which after purifica- tion, melt at 38°, and are identical with methylnitramine, as was proved by comparing the compound with a specimen kindly presented to us by Prof. Francuimont. The yield, however, was very small. If the finely powdered nitramine, m. p. 127°, is treated with 20 °/, methylalcoholic ammonia this becomes intensely red, the mass gets warm and after a few hours the reaction is complete, and a large amount of picramide has formed which is removed by filtration. The aleoholie solution is distilled in vacuo, the residue treated with dilute sulphuric acid and, after removal of a yellowish byeproduct by filtration, the liquid is agitated with ether. On evaporation of the ether, crystals of methylnitramine were obtained. In this reaction also, the yield was not large, amounting to only 15 °/, of the theoretical quantity. With ethyl-alcoholie ammonia a similar result was obtained, whereas an experiment in which ammonia was passed into a solution 1) Rec. d. Trav. chim. d. Pays-Bas, Il. (1883) p. 115. 2) Ib. VIII (1889) p. 281. 3) Ib. VII (1888) p. 354. ( 706 ) of the nitramine in benzene gave results which were still less favourable. One of us had noticed previously that among the aromatic amines which generally react on an alcoholic solution of the nitramine quite as readily as on pieryl chloride, p-toluidine in particular gives a beautifully crystallised p-toluylpicramide m. p. 166°?) whilst the alcoholic solution contains only comparatively few, not very dark coloured byeproducts. In an experiment in which 35 grams of the nitramine were heated on the waterbath with an equal weight of p-toluidine and 100 ce. of 96°/, alcohol, a fairly violent reaction set in after some time. The heating was continued for 5 hours and, after the picramide derivative had been removed by filtration, the alcohol was distilled off and the residue extracted with dilute sulphuric acid. The liquid filtered off from the toluidine sulphate was shaken with ether. On evaporation of the ether a still yellow coloured liquid product was left which on being inoculated with a erystal of methyl- nitramine became crystalline and after having stood for some time over sulphuric acid weighed 7 grams. On pressing between filter paper light yellow crystals were obtained which after being sublimed in vacuo (a treatment which methylnitramine stands very well) melted at 38°. On mixing the same with a preparation con- sisting of pure nitramine the melting point was not affected. p-Toluidine appears, therefore, to be a suitable means for readily procuring in a short time methylnitramine from s-trinitrophenyl- nitramine. We are continuing our investigations with different amines and also with other nitrated aromatic nitramines, and will state the results more elaborately in the ‘Recueil’. Org. Chem. Lab. of the University Utrecht. Physics. — “ Wave-lengths of formerly observed emission and ab- sorption bands in the infra-red spectrum.’ By Prof. W. H. JULIUS. If in the infra-red spectrum, as formed by means of a rock-salt prism, the positions of emission or absorption bands have been care- fully determined, the corresponding wave-lengths still are uncertain by an amount which, in a considerable part of the spectrum, is greater than the probable error of those determinations, because the 1) We now obtained this substance in two modifications, one coloured dark red and the other orange. ( 707 ) dispersion curve of rock-salt is not yet known with sufficient exactness. Mr. W. J. H. Morr *) has lately compared with each other the dispersion curves that have been calculated according to KerreLER’s formula with two sets of constants, one given by Rugens °), the other by LANGLEY®). LANGLEY's results held for a temperature of 20°; the numbers given by RuBreNs were corrected by Mr. Morr so as to apply to the same temperature. While coinciding in the visible spectrum, the two dispersion curves appeared to diverge very sensibly in the entire infra-red region, the wave-lengths correspond- ing to given indices of refraction being smaller with RvBeNs’ than with LANGLEY’s constants. At 2=1,5u e.g. the difference amounts to 0,028 u; it increases unto 0,062 u (at 2 == 3u) and then decreases to 0.032 u (at 2= 8,5 u). If, on the other hand, the indices of refrac- tion, which according to LANGLEy’s and according to Rupens’ formula belong to rays of given wave-lengths, be compared with each other, the difference appears to be rather constant between 2—=-4 and 2 =8,3 u, namely 1,5 units of the dk decimal of the index, and to increase from O to 1.5 similar units in the region between 0.6 u and 4 u. The apparatus, nowadays available for the investigation of the infra-red, admit of determining the position of sharp maxima or minima of radiation with an accuracy, going a good deal farther than 1,5 units of the 4'" decimal of the index. When between 1887 and 1891 I investigated several infra-red emission and absorption spectra, our knowledge of the dispersion of rock-salt was restrained to the outcome of LANGLEy’s first determi- nations *), which extended only as far as 5,3u. As a great part of my work bore upon longer waves, I published my results in the form given by direct observation, viz, as galvanometer deflections and corresponding angles of minimum deviation, reduced to the temperature 10°. The refracting angle of the prism being also recorded, the indices of refraction of rock-salt for waves, corresponding to the observed maxima, were thus implicitly given. In order to obtain a rough estimate of the wave-lengths, I had extended LaNnGLry’s dispersion curve in a straight line, though under strict reservation. The wave-lengths as read on this lengthened 1) W.J.H. Mott, Onderzoek van ultra-roode spectra. Dissertation, Utrecht, 1907. 2?) H. Rupens, Wied. Ann. 60, 724; 61, 224; 1897. Cf. also Kayser, Handbuch der Spectroscopie I, 371, 1900. 5) S. P. Lanerey, Ann. Astroph. Obs. of the Smiths. Inst. I, 1900. 4) S. P. Lanetey, Phil. Mag., Aug. 1886. (708) curve, to which I myself assigned little weight *), have found their way to some text-books ®), where they unfortunately appear as the results of my investigation, with the incidental remark that they are incorrect, as founded on a false extrapolation. It is clear, however, that this incorrectness has nothing to do with the accuracy with which the position of the bands in the prismatic spectrum has been determined. Now I have reason to believe, that the spectrometric and heat-measuring apparatus used in that research were not less valid than those employed by many later observers of infra-red spectra (Donatu, Puccianti, Ike, COBLENTZ, NicHoLs, RuBENs and ASCHKINASS and others), so that the results still retain their value as a first contribution to our knowledge of the examined spectra. I therefore thought it suitable to republish the principal results obtained at that time ®, but now to mention the indices of refraction for the maxima of emission and absorption, as following directly from my observations, and to add the wave-lengths, as derived from the more recent dispersion curves of RuBeNs and of LANGLEY. The positions in the infra-red were determined in my work with respect to the place of the D-lines of a Bunsen flame coloured with chloride of sodium. But the latter were too faint to be observed with the bolometer; and the transition from the visual observation of the D-lines to the bolometric observation of infra-red radiations caused an uncertainty in the determination of the relative positions, which was still increased through the necessity of displacing the bolometer along the optical axis of the rock-salt lens according to its different focus for visible and invisible rays. It was chiefly in the part of 1) Cf. ,,Bolometrisch onderzoek van absorptie-spectra”’, Verhandelingen der Kon. Akad. v. W. te Amsterdam, Vol I, N°. 1, p. 8 (1892), or the German translation in: Verhandl. des Vereins zur Beförderung des Gewerbfleisses, 1893, p. 235, where I have clearly stated that | considered the extrapolation of Laneey’s dispersion curve as quite uncertain, and that in the tables the direct data of observation (angles of minimum deviation) were given, because I did not like to have my results inseparably connected with a possible incorrectness of the dispersion curve. The passage in question seems not to have been noticed by W. W. CoBrentz, for in his excellent work ‘Investigations of Infra-red Spectra”, published by the Carnegie Inst. of Washington, 1905, he says on p. 135, after alluding to LANGLEY's extrapolation of the dispersion curve in a straight line: ‘Junius, with apparently less hesitation, has applied this extrapolation to his work”. 2) WiNKELMANN, Handbuch der Physik; Kayser, Handbuch der Spectroscopie ; Cuwotson, Lehrbuch der Physik. 3) Recherches bolométriques dans le spectre infra-rouge. Arch. neérl. 22, p. 310—383 (1888). Die Licht- und Wärmestrahlung verbrannter Gase, Berlin, Simion. 1890. Bolometrisch onderzoek van absorptiespectra, 1. c. ( 709 ) the investigation, deseribed on p. 69 of “Die Licht und Wärme- strahlung verbrannter Gase” that many pains were taken to reduce this source of error. There the CO,-maximum of the Bunsen flame was found at minimum deviation 38°54'20", the refracting angle of the prism being 59°53'20" and the temperature 10°. From this follows n= 1,52103. Had the temperature been 20°, then the devia- tion would have been found smaller by 1/50”, giving for the index of refraction: 2 = 1.52069. If. we suppose this value to be exact, then the angles of minimum deviation given in my first paper in Arch. néerl. 22, and on pages 4+7—68 of “Die Licht- und Wärmestrahlung” are too small by nearly 3', owing to an instrumental error. In “Bolometrisch onder. zoek van absorptiespectra’ the deviation of the CO,-maximum has been found 38°5240" instead of 38°54'20"; 1' of this difference results from the fact that the refracting angle of the prism, then in use, was smaller than that of the other one by 1'; only the remaining 40” were owing to an instrumental error. I have now applied the corrections resulting from this re-exami- nation, and calculated the indices of refraction for 20°, the tempe- rature to which the dispersion curves as compared by Mr. Morr also refer. In finding the wave-lengths corresponding to the indices, advantage has been taken of elaborate tables, prepared by Dr. Morr for a research of his own, and which he was kind enough to put at my disposal. (2109 Emission-spectrum | Indices of refraction Wave-lengths according to | | the dispersion curve | for the maxima Intensity ') of: | (Temp. 20 ) | of RUBENS | of LANGLEY BUNSEN-flame .... - | 1.5268 | 1.905 1.958 0.5 1.507 HO | 2769 | 2.831 3—5 1.52069 CO, 4.410 4.462 10 Flameofcarbonmonoxyde, 1.52445 CO, 2.883 2.947 is or of cyanogen 1.52069 CO, 4.410 4.462 10 Hydrogen flame... . 1.5247 H,O 2.77 2.83 10 1.5176 5. AA 5.46 Luminous gasflame. . . 4.5270 Ee 1.84 4 89 2 4.5247. H,O 2.77 2.83 2) $:5207:..° COs 4.44 4.46 2) Hydrog. burning inchlor.) 1.5226 HCI | 3.68 3.74 Flame of sulfur... . 1.5093 SO, 7.49 7.52 Flame of carb. disulphide} 1.5247 2.77 2.83 1 1,504 CO: L.A 4 46 10 1.5125 COS(2)| 6.76 6.80 3—0 2) 1.5093 SO, 7.49 7.52 9-3 % Absorption-spectrum of: C (diamond)... . . 4.5938 3.18 3.4 1.5202 4.58 4.63 10 1.5183 5.20 5.25 1.5088 etc. *) 7.59 7.62 10 BROER ee 1.5287 1.44 1.43 4 4.5265 2.01 2.06 4 4.5236 3.25 3.31 95 1.5194 4.85 4.90 1.5146 6.24 6.28 1) In each spectrum the intensity of the highest maximum is indicated by 10. The letter s following an intensity figure means, that the band is rather sharp. 2) The relative intensity of these bands varies much with the place in the flame. 3) The addition “etc” behind an index of refraction indicates, that the band marks the beginning of an extensive region of strong absorption. ( 714 ) Absorption-spectrum of: Indices of refraction Wave-lenghts according to for the maxima (Temp. 20°) 1 1 eee a: en ien in ie SS ee ee end „5203 „5129 the dispersion curve of RUBENS hi LANGLEY 4.55 | 4.60 6.67 | 6.71 ike eee ee ene be ef 5.68 Denen VRB 8.36 8.39 8.90 893 | 9.44 * 9.44 | 10.28 NE | 3.88 3.04 7.72 7.75 8.73 8.76 1025 | 41098 5.53 5.57 6.03 6.07 8.19 99 9.02 9.05 9.73 9.76 | 3.34 3.40 5.50 5.54 8.19 8.92 9.02 9.05 9.73 9.76 6.47 6.54 8.19 8.29 10.28 10.31 3.34 3.40 6.62 6.66 6.99 7.03 8.19 8.99 9.62 9.65 | Intensity 6 Ss Co 10 (A35) . Indices of rashonden gths according to} _ dede for the maxima __ the dispersion curve Intensity of: (Temp. 20°) of RuBENS | of LANGLEY Spb) ee reg Re emer 1 5235 3.30 3.36 3 s 1.5207 4.40 445 3 s 1.5164 5.76 5.80 4 s 1 5116 6.97 7 Ot 3 1.5083 8.19 8.22 7 15008 | 888 || 8.86 10 4.4992 9.44 9.44 6 CATS 03, SEREN | 1.5259 2.95 2.31 1 pee MES Rl Sl is 4.5211 EEN |e sah eal 1 | 1.5173 | 5.50 es) eee 6 4.5128 6 69 6.73 40 s | 1.5107 7.18 7.21 3 s | 1.5088 1E ie 1 1.5060 8.15 8.18 2 425039 8.56 8.59 7 (Cla: 2b ten | 1.5259 2.25 2.31 2 | 1.5230 8.50, bl 57 s 15154 6.03 || 6.07 1 1.5118 6.93 6.96 10 st) 1.5097 7.40 7.43 1 5068 7.99 8.02 | 4 4.5032 8.69 8.72 6 s 1.4980 9.63 9.66 5 1.4942 Ne 40:28: hap SI 5 OE Pen | 459 | 25 || 2.3 1 | 1.5229 3.56 3.62 10 1 5194 OECD OO 2s 1.5145 etc. | 6.97 6.31 | 40 GRON eertse 4.5259 2.25 2.31 1 4 5229 3.56 3.62 10 1.5183 5 20 5.25 2 s 1.5154 6.03 6.07 1.5126 etc. 6.74 6.78 10 1) Sharply limited only toward the smaller wave-lengths. EEE Absorption-spectrum of: for the maxima (Temp. 20°) (24: 46); ee ee 4.5230 4.5152 1.5126 etc. C,H,OH (normal)... . 4:.5230 4.5162 4.5126 etc. CHOH (iso). jee 1.5230 1.5192 4.5154 1.5126 etc. A a ee sae 1.5230 1.5126 etc. 4.5045 etc. >= = Oo — ler) bo of RUBENS | of LANGLEY Indices of refraction Wave-lengths according to the dispersion curve 3.51 6.09 6.74 3.51 5.81 6.74 3.51 4.92 6.03 6 74 3.01 4.92 6.03 3.57 Intensity 2 8s 3 4 (74 Physics. — “A hypothesis relating to the origin of RÖNTGeN-rays.” By Prof. C. H. Winp. W. Wien’) has measured the energy of RÖNTGEN-rays, converted into heat in a bolometer or in a thermo-element, and has compared it with that of the cathode-rays, likewise converted — with exception of the small fraction transformed in energy of R.-rays — into heat in the anti-cathode. He finds for the proportion of the total quantities of energy of the two kinds of rays sa Ae er Ex Supposing that the R.-rays are the radiation of energy, emitted by cathode-ray electrons being brought to rest, and that this stoppage may be considered as a continually decreasing motion, he proceeds with the aid of the theory of M. ABRAHAM to deduce the duration of the stoppage and from it the thickness of the R-waves. For the latter he finds des d,lor, 1O= 8 cor, Results of the same order of magnitude have afterwards been attained by Epna CARTER*) in an investigation, also made at the laboratory directed by Wien. These results do not very well agree with the values, derived by Haca and myself for the wave-length of R.-rays from diffraction- experiments : A= 270 to 12 . 19—lem *) and — 160 , 120 , 50 . 10-10 em. 5) If the R.-rays have to be considered as disturbances in ether of the single pulse character assumed by Wien in accordance with the current conception, the same numbers must be divided by Te 2 24°) in order to represent the corresponding values of the thickness of the pulse-waves, which consequently become B,=110 to 5 20-2. em, PB, =~ 64, .48,- 20°. 1010 em: 1) W. Wien. Witiners Festschrift, Leipzig, 1905; Ann. d. Ph. 18, p. 991, 1905. 2) L. ce. p. 996. The number is doubled here, on account of the remark made regarding it on page 1000. 3) E. Garter. Ann. d. Ph. 21, p. 955, 1906. 4) H. Haca and C. H. Winn. These Proc. I. p. 426. 5) Id. Ibid. V. p. 254. 6) CG. H. Winn. Physik. Zschr. 2, p. 96. Fussnote 2), 1901. ( 715 ) Wirn’s experiments would have led to results more in keeping with the diffraction experiments, if the values found for the energy of the R.-rays had been 20 to 100 times smaller. The difference is too great to ascribe it to errors of observation. We must rather think of fundamental errors in the method of observation or of a viciousness in our conceptions concerning the mechanism of the phenomena. As for the method of observation Wien himself pointed out’) the possibility that the quantity of heat, generated in the bolometer or in the thermo-element, should not be to its full amount converted energy of R.-rays, but partly also — perhaps even for the greater part — converted atom-energy, liberated by a, say, catalytic action of the R.-rays. J. D. v. p. Waars Jr.*) suggests the additional idea that the electrons are not generally stopped at once by a simple uniform decrease of velocity, but will mostly, by their interacting with the particles of the anti-cathode, before being brought to rest move for some time amidst the latter in rapidly changing directions with great - velocities, sending out a new R.-pulse at every change of motion. Starting from this idea we could, indeed, expect from each electron a much greater contribution to the energy of radiation than in the theory accepted by Wien and find the results of WieN’s energy- measurements in better agreement with those of the diffraction- experiments. Nevertheless it seems to me that by the side of this another idea deserves our attention, which might be more in keeping with the properties of cathode-rays as far as known. It would be this, that not simply the cathode-ray electrons, but in combination with these the atoms of the anti-cathode are the principal centres of emission of R.-rays. It should be imagined, that the electrons, arriving at the anticathode with their immense velocities, are not, generally, thrown into an other direction by the atoms, but will for the greater part pass straight through them, and even, in doing so, will mostly not suffer any persisting decrease of velocity. This idea is by no means a new one. It has been worked out by Lenarp*), who sees in it the best explanation for the laws of absorption of the cathode-rays. In very few cases only it will happen that an electron, when piercing an atom, gets imprisoned 1) W. Wien. Drudes Ann. d. Ph. 18, p. 1005, 1905; cf. also E. Carrer. Ann. ad. Phee2i,-p.). 957; 1906. 2 J. D. v. p. Waats Jr. Ann. d. Ph. 22. p. 603. 1907. 3) P. Lenarp. Drudes Ann. d. Ph. 12, p. 734, 1903. 49 Proceedings Royal Acad, Amsterdam. Vol, IX, ( 7165 or changes its direction considerably *) in a centre of exceedingly strong electromagnetic action; in the great majority of cases it will, by the abundance of vacant space in the interior of the atom *), fly across it without experiencing a considerable decrease of velocity. In this way the greater part of the electrons will pierce thousands or tens of thousands of atoms before being stopped, and we find easily explained the great penetrating power of the cathode-rays, which may still in appreciable quantity pass through a layer of aluminium 10 u°) thick or a layer of atmospheric air, some em thick *). If we consider the values given by the diffraction-experiments for the order of magnitude of the thickness of R.-waves as correc}, it follows from Wiens experiments — apart from a possible catalytic action of the R.-rays — that the radiation of the cathode-ray corpuscles, by the simple fact of their stoppage, could account only for something like el or me of the whole energy of the R.-rays. Conse- 20 100 ë quently for by far the greater part this energy must, if LENARD's views may be accepted, have a different origin. What this can be, is obvious. The atoms namely will by no means remain undisturbed during the sudden passage of an electron. Themselves probably con- sisting of negative and positive corpuscles, they will see their electro- magnetic fields during the passage altogether altered and at the same time will no doubt send out a pulse or wave of disturbance °) into the surrounding ether. About the character or shape of these pulses, which moreover may vary from one case to an other, we can, without making any more definite assumptions as to the structnre of the atom, say little; but there is one important point, in which all these pulses will be to a certain degree similar, viz. their duration. 1) Together with the expulsion of electrons originally belonging to the atom, which will often occur at the same time, these changes of direction could very well account for the diffusion of the cathode-rays according to LeNARD. 2) Lenarp calculates (Drudes Ann. d. Ph. 12, p. 739, 1903) that only LO—9 of the volume of an atom is occupied by the “dynamids”, of which he considers it to consist. 3) Lenarnp. Wied. Ann. 51, p. 233, 1894. 4) Id., Ibid., p. 252. 5) Lenarp expresses himself (“Ueber Kathodenstrahlen”, Nobel-vorlesung, p. 37, Leipzig 1906) as follows: “Das durchquerende Strahlenquant’” — the electron — “wird vermöge der abstossenden Kräfte, welche es auf die anderen, dem Atom eigenen, negativen Quanten ausübt, eine gewaltige Störung innerhalb des Atoms hervorbringen kénnen’’, and then continues thus: ‘und als Folge dieser Störung kann ein dem Atom gehöriges Quant aus ihm hinausgeschleudert werden (sekundäre Kathodenstrahlung)”’; but he does not speak of a radiation emitted by the atom. (717 ) The latter will be, if @ represents the diameter of an atom and v the velocity of the electron, which is piercing it, something like a . . . , causing the wave emitted to be of a thick- v (rather smaller than) a . ness of something like (rather smaller than) ec —, c being the velo- ()) city of light in ether. By putting a=10-* and v= 10", we get a . . by this way for c — 3.10—-8, a number which only slightly exceeds v the order of magnitude of the values of 8 (p. 714), derived from diffraction experiments. It might therefore be possible, that the waves of disturbance in question should be identical with the Rünt- gen rays. As by this theory a single electron would disturb some thousands or tens of thousands of atoms, every atom, being traversed by an 1 electron, need only send out something like 100 of the quantity of energy emitted by an electron itself in its total stoppage, in order to account for the relatively large amount of energy found by Winn in the R-rays. That such proportions should exist, seems to me not impossible at all. The views presented here as to the origin of the R.-rays bestow a new and great importance on the ‘‘wave-length” of these rays, as they intimately connect this measurable quantity with the dimensions of the atoms. Whether there really exists such a close connection, could perhaps be experimentally put to the test by diffraction experiments with anticathodes made from different materials. More generally it might be expected that experiments of this kind would throw some new light upon the structure of atoms, and also of molecules or molecule aggregates. In such experiments it would certainly have a peculiar interest to use erystals as anticathodes, as perhaps the regular structure of these bodies could manifest itself both in rather sharply defined wave-lengths of the R.-rays emitted by them as in a polarisation of these rays. The question, whether R.-rays should or should not be expected to show total or partial polarisation, may be treated on the basis of the above hypothesis, as soon as this be supplemented by definite suppositions about the structure of the atom. The relation that, according to our views, should have to exist between the wave-length of R.-rays and the velocity of the cathode- rays, is of course liable to rather direct experimental verification. 49* ( 143} Two further questions connected with those views and perhaps liable also to be answered by way of experiment, are these: 1. whether the air molecules on the outside of the aluminium window of Lunarp emit R.-rays in appreciable quantity ; 2. whether the y-rays of a radio-active substance, except by the substance itself, are to a considerable extent emitted also by the atoms of air in its neighbourhood on their being pierced by the electrons constituting the g-rays. Physics. — “On the motion of a metal wire through a piece of ice.” By Dr. J. H. Mrersure. (Communicated by Prof. H. A. Lorentz). (Communicated in the meeting of January 26, 1907). During the last and the preceding winter I made some measurements with a purpose of testing the formulae, expressing the velocity of descent of a metal wire through a block of ice, which Mr. L.S. OrnstEIN had derived from the theory of regelation *). In my experiments the metal wire was fastened at both ends to DL. S. Ornstein. These Proc. VIII, p. 653. ( 719 ) the legs A and D of an iron frame, which, in order to secure greater rigidity, had been cut from an iron plate. In the first measure- ments the downward displacements of the wire were observed by means of a small reading telescope, turning round a horizontal and a vertical axis, and were determined on a measuring rod, mounted at the side of the frame. The breadth of the ice-block was also read on a horizontal measuring rod. In the later experiments a catheto- meter was used, placed at my disposal by the professors of the Technical University at Delft. | wish to express here my sincere thanks to these gentlemen, especially to Prof. pr Haas. The fall of the wire was always derived from the change in the difference of level between the top of the wire and the upper edge of a small bubble, existing somewhere in the interior of the ice. Every ten minutes or, when the descent was quicker, every five minutes, the difference of level was measured in order to ascertain whether the fall was regular. Each experiment lasted 20 to 40 minutes. The ice used was artificial commercial ice. From a larger block a clear smaller one was sawn out, in which some bubbles should be present to serve as marks. The faces were melted flat by pressing them against a metal plate, so that errors, caused by irregular refraction, were avoided. Heat conduction along the wire was prevented by hanging small pieces of ice on the wire on both sides of the block. Still small grooves were occasionally formed when the descent was slow. The experiments were made with wires of steel, german silver and silver. The thickness of the wires was measured by means of caliber compasses, giving results accurate to 0.01 mm. The thickness was 0.5, 0.4 and 0.8 mm. Deviations from these numbers, amount- ing to some hundredths of a millimetre, were occasionally found. For the case, realised in my experiments, in which the two straight ends of the wire make a certain angle 2e with each other, formula (IIIa) of Mr. Ornsrern’s paper *) holds: 2aCP i= d, sin a - (1) in which v represents the velocity of descent of the wire, P the total weight and d the breadth of the ice-block. C is a constant. The value of this constant I calculated by formula (1) from the values of v, found in my experiments. The results are summarised in the following table: EG: Steel wire. (in grammes). German silver. Silver wire. i 455 155 1255 2160 2205 5160 455 755 1255 ( 720 ) Diameter 0.5 rea 0.0162 0.0151 0.0172 0.0185 0.0169 0.019 diameter 0.4 0.030 0.029 0.029 diameter 0.3 0.043 0.042 diameter 0.5 0.0134 0.0119 0.0143 0.0172 diameter 0.4 0.0196 0.0204 0.0208 0.0255 diameter 0.3 0.0306 0.0348 0.0398 diameter 0.5 0.0207 0.0255 diameter 0.4 0.0367 0.0384 0.0392 0.0404 diameter 0.3 0.0347 0.0467 mM. | mM | mM. | mM. \ mM. \ mM. | \ mM. mM. mM. | average of (. 0.017 0.029 0.0425 0.014 0.022 0.055 0.023 0.039 0.041 ( 721 ) The quantity C is not expressed here in C.G.S. units, since the dimensions have been taken in millimetres, the velocities in milli- metres per minute and the forces in grammes. In order to reduce them to C.G.S. units, the value of Chas to be multiplied by 170 >< 105. The values given in the table are averages of several measurements. In order to show the deviations of different measurements, made with the same weight, I give here an arbitrarily chosen set of separate measurements. German silver wire, diameter 0.4 mm. Number of the experiment iP p oO, 2a C averages. 8 1036. LOL, BROS 42 0:0a68 Ee KOSO ft 37.4 42 0.0393 | 0.0399 14 £036) LA “BRA TAO OEL \ 17 1056. 1473, 29.5. “60? AO: 0308 87 1255. 0:90 52.3 50°. 0.0401 112 12555, 1.09) DAA DI - 0.0428 0.0404 115 12054 07756 66.46.53) O0ean The value of C is calculated by Mr. Ornstein in formula (1) of his paper. He finds dt hk —4a@p(k, —k (=) + nl An | dp 0 k, =P d/R (4, ra k,) CZ sige want) x RWS; Here £,, &, and &, are the coefficients of heat conductivity respec- : dt tively of the wire, of water and the ice, (=) is the rise of the ap 5 melting temperature by pressure, measured at the melting tempera- ture, W is the latent heat of melting ice, S; the specific gravity of ice, A the radius of the wire and d the thickness of the layer of water. Now the value of C cannot be calculated by this formula, since the quantity d is unknown. But besides the equation (I) Mr. Ornstein gives in his formula (II*) an expression, found by a hydrodynamical reasoning, in which the quantity d likewise occurs. This relation is‘): Sw ZaP ax VvE | — EURE ak aa 5 S; 12 2ud,sina\R re ; Sw. ') In Mr. Ornstern’s paper this formula is given without the factor since = U this latter has no perceptible influence. ( 70%) Here S,, is the specific gravity of water at 0°, u the viscosity coefficient. By equalising (1) and (3) we find: C aa Sw 1 d ke 4 TET d and we should now have to eliminate z between (2) and (4). In order to perform this elimination we simplify (2). We consider the form d in (2) between the brackets [| | and keep in mind that = small, that #, is very much greater than 4, and that 4, may be neglected with respect to the first term (which amounts to neglecting the conduction of heat through the ice). We may then write: he d/p (tk, — &,) Ling k, vn k 7k + YR, kk) “he + RR dt k, dp k, Prat d/ ii + YR is very 1 L- dp k, Then we have C=] x R? WS; If we put dt c Y nr 1 (G)=—74 X10, S= 0.9167, W= 79.2, dp), C becomes 1 hy CBR ae +e SS In (4) we substitute S,=1, S= 0.9167, «= 00181, a? ¢ = 1.600 (5) he See Coane A Equalising the two values of C we have: de 1 k, EN e= ln R Di 1 d R d ae GD ED RI TAR meen Rr then k, k, or 2 ( 723 ) d Fi can easily be found by a tentative method, 1 From this equation when &,, & CGS-units), ‚and A are given. In the different cases we find (in d Steel wire k==i04166 4, == 0001S “Res= 0.025 RS 0.00166 i/ kh, = 0.166 k= 0.0015 R= 0.020 5 — 0.00190 b, = 0166 4, = 0.0015 R=0.015 = = 0.00229 German silver wire k, = 0070 k, = 0.0015 R—0.025 —— 0.00128 I k,=0.070 k,= 0.0015 RR 0.020 == 0.00149 d k, = 0.070 k,= 0.0015 R=0.015 — = 0.00179 d Silver wire b= 1.50 k= 0.0015 R= 0.025 | = 0.00239 : d k= 1.50 k= 0.0015 R=0.020 = = 0.00279 d b= 150 k= 0.0015 K=0.015 = = 0.00818 Cis then found by substitution in (5). These values are given below, together with the values found by experiment, but now expressed in CGS-units. Calculated Found Steel wire His 0028 torch 29 x 10—'0 ge 0.020 Te x ig 49 > 10—10 f= 05 1923 10-0 72 X 10-19 German silver wire R= 0.025 34 & 10—10 24 X 10—?0 R= 0,020 53 Xx 10-10 37 & 10-10 R=0015 91 x 10-10 59 & 10-10 Silver wire R= 0.025 218 «K 10-10 46 >x 10-10 R= 0.020 347 X 10-10 66 1010 = 0,085 519 X 10—10 10.5% 100 The agreement must be called bad for the silver wire, satisfactory for the german silver wire. It may be called satisfactory, since different circumstances may be mentioned which make us expect a too small value. Leaving aside the great uncertainty in the values of the heat conductivities of metals, to which we cannot here ascribe the bad agreement, since we do not know in which direction this (724 ) will influence the result), the following causes may be mentioned. 1. The roughness of the wires. Already Mr. ORNsTrIN pointed this out. If the wire is not entirely smooth, the hydrodynamical deductions are uncertain and hence also formula (3). In order to ascertain the influence of this roughness | made some experiments with a steel wire that had for a moment been scoured with fine sand-paper in the direction of its length. Macroscopically no result of this manipulation could be discovered on the wire. Yet the effect proved considerable, for the following results were found : ia C \ 455 0,009 | steel wire, diameter 0,5 m.M. {1255 0,011 ;average 0.011 m.M. | 2205 0,014 | steel wire, diameter 0,3 m.M. 1255 0,028 So we find a diminution of about 40°/, in the value of C. After having observed this influence I tried to obtain smooth wires, but unsuccessfully ; all the wires that were used in the experiments showed under the microscope numberless grooves in the direction of their length and of a breadth that might be estimated at somewhat less than 0.01 mm. : d Since it is easily deduced from the calculated values of R that the thickness of the layer of water increases with the size of the radius of the wire and since the influence of the roughness of the wire will be smaller with a greater thickness of this layer of water, I have still made some measurements with a thicker steel wire of 0,87 mm. diameter and heavier weights. The result was: Js C C (in C.G.S. units) 25200 0,00805 | 25200 0,00822 | 5200 0,00667 HAES er while calculation gives d bk == 05166, 4, = 0 00ON Ri EI ee ae Cl 0,0081 13.8 < 10-1 The agreement is now better indeed; the value found is half the calculated one, while with the thinner steel wires it was slightly more than a third. 1) The values given by F. Kontrauscn (Lehrbuch der praktischen Physik 10 Auflage 1905), steel k = 0.06 to 0.12 and silver k—=1.01, would give a much better agreement. ( 725 ) 2. In the deduetion of formula (1) it was assumed that within the layer of water the relation 0 0 de? | dy? Es! holds. This relation, however, holds for a body at rest. Here, on the other hand, we have to deal with a streaming liquid, in which case the following formula holds: Oe. On Sw Ot Ot Ba eee k, (eas +75): Here o is the specific heat of the liquid, « and » the velocity components in the X- and Y-directions. If we use this formula we take into account that the heat, conducted through the wire, does not entirely serve for melting the ice, but that it is partly conveyed upwards again by the streaming liquid. This also must result in a diminution of the velocity of descent. Prof. Lorentz informed me, however, that it can be shown that this influence must be regarded as a quantity of the second order, so that the differences cannot be explained in this way. 3. If the temperature in the interior of the block of ice is not exactly 0°, but lower, the velocity of descent will also become smaller. But I observed no phenomena which point to a lower temperature in the interior. Blocks of ice that had been kept for 24 hours in a space above O° gave the same results as blocks that had just been received. Moreover the wires as a rule went down at a distance of only a few millimetres from one of the faces of the block, and in some experiments they even came out of the block by melting of that face. Yet in the last moment, before the wire came out, no acceleration of the descent was observed. Nor does theory support such an explanation. Prof. Lorenz informed me that when the surface of a ball of ice of 3 centimeters diameter and at a temperature of — 2°, is raised to 0° and kept at this temperature, it may be shown that in less than an hour the temperature at the centre has risen to — 0.01°. 4. Another important influence on the velocity of descent is found in the fact that it is possible that not all the ice, melting at the lower side of the wire, freezes again exactly at the upper side, but that this water perhaps flows off laterally. It is clear that this must have a great influence since then the heat, necessary for the melting, is furnished by conduction through the ice. Already J. Thomson ( Bey BorromLey ') showed that the lateral flow of water causes a great retardation. The experiments now showed that this lateral flow really exists. For even when the ice was perfectly clear, in the places where the wire had passed through it various small bubbles were observed. Consequently not all the ice had been re-formed which had been there. In this respect I also mention a curious change, found in the values of C: these values rise with the weight. This is very con- spicuous with the silver and german silver wires, but also with the steel wires it exists, especially with the thick one of 0.87 mm. Accordingly it was often seen that the bubbles on the path of the wire were more numerous with small than with heavy weights. This became particularly clear in experiments in which, during one descent, first a heavy and then a small weight was used, With the smaller weight more water flows off laterally. I still made several experiments in which the wire was pulled upwards through the ice, hoping to prevent this lateral flow. The result was not the expected one, for bubbles also appeared and the values, found for C, were even somewhat smaller than in the former case. In regard to this question it would be desirable to investigate the descent of a whole body, e.g. of an iron ball, through perfectly clear ice. In my opinion this lateral flow is the chief reason why theory and observation disagree. It also explains why with the silver wires larger differences were found than with the german silver and the steel wires. For if the heat is only partly furnished by the freezing process above the wire and if the rest has to be furnished by con- duction through the ice, it seems to be of little consequence whether the wire be a good conductor of heat. 5. Ice is a crystalline substance. This also may have its effect. Perhaps the melting point is not the same at the different faces of the erystals which the wire touches. Though this influence may exist, we cannot say in which direction it would modify the result. In order to find out whether such an influence makes itself felt, I made the wire pass several times through the same block of ice in three mutually perpendicular directions. But no perceptible diffe- rence was found. As the general result of the experiments I think we may state, that they indicate that the regelation theory will be found capable of explaining the phenomena not only qualitatively but also quanti- tatively. 1) Pogg. Ann. 148, p. 492, 1871. ( 707) Physics. — “Contribution to the theory of binary mictures, HL’, by Prof. J. D. van DER Waats. (Continued, see p. 621). Not to suspend too long the description of the course of the qg-lines 2 , y in the case that the locus — = 0 exists, we shall postpone the deter- adt mination of the temperature at which this loeus has disappeared, and the inquiry into the value of w and v for the point at which it disappears — and proceed to indicate the modification in the course of the q-lines which is the consequence of its existence. dp 9 dv an. From the value of — = — — follows that when a q-line dag dp dadv 2 passes through the curve a = 0, it has a direction parallel to the U Fig. 3. ( 728 ) : i … Gr x-axis in such a point of intersection. So a q-line meeting Ei ax” will be twice directed parallel to the z-axis, and have a shape as : dy dp represented in fig. 3 — at least as long as the curve = adv des does not occur. Such a shape may, therefore, be found for the g-lines, in the case that the second component has a higher value of 6, and lower value of 7% —, and such a shape will certainly present itself in the case mentioned when the temperature is low enough. Then there is a group of g-lines, for which maximum volume, and minimum volume is found. The outmost line on one side of this group of g-lines, viz. that for which g possesses the highest value, is that for which maximum and minimum volume have coincided, 2 wp ; aes and which touches the curve — = 0 in the point, in which this curve v itself has the smallest volume. The other outmost line of this group of q-lines, viz. that for which g possesses the smallest value, is that for which again maximum and minimum volume have coincided, and dp which also touches the curve Pa = 0, but in that point in which ve this curve itself has its largest volume. So for these two points of — aw . ; contact the equation ——~-— 0 holds. These two points of contact are, Wl bes dy dy therefore, found by examining where the curves —— = 0 and — == 0 an at intersect. This last locus appears to be independent of the temperature, d*a 4 as we may put — equal to 0. We find from the equation on p. 638 ax db\* d°b dp 12e dx de ==) 2 SAL | da? xv? (lx)? v—b)* (v—6)? d? ; 3 If we neglect Fe we find from oS db dz & 3 12 v—b 22° (1—«)? The locus = 0 occurs, therefore, only in the left side of the Ak figure or for values of « below 4. The line « = '/, is an asymptote for this curve, and only at infinite volume this value of z is reached. ( 729 ) 3 uw And as for x=0O also v—b must be — 0, the curve —— = 0 starts an feta from the same point from which all the gq-lines start. If — should Ae not be equal to 0, we have ground for putting this quantity positive (Cont. II, p. 21), and we arrive at the same result for the initial 3 point and the final point of the curve En ==.) an” dp So the points of the curve —- == 0, where tangents may be Av drawn to it parallel to the x-axis, lie certainly at values of « smal- ler than 4, and accordingly the two outer ones of the group of the g-lines with maximum and e minimum volume have their hori- : zontal tangents also in the left side of the figure. The g-line with the highest value of q at lower value of « than that with the lowest value. This is repre- sented in fig. 4. We notice at the same time that the points in which a q-line dw Fig. 4. touches the curve i ie ae at points of inflection for such a g-line, just as this is the case with . . a? ml the p-lines when a p-line touches the curve sats From dv dw as da Py (dv : ee 0, dadv \ dx g a Py dv dw id 4 dep (dv db) ae A Bir age eee dadv dax*, dede? \ dx}, dx*dv \ de), de? dv l uw In the points mentioned é ee because —~== 0, and at the q follows : and Ae” ( 730 ) ? wy d*v : ; same time aa = 0. Hence =: = 0, which appears also immedia- He U q tely from the figure. 2 Within the curve ae 0 every g-line that intersects it, has also av a point of inflection, because the latter must pass from minimum volume to maximum volume in its course. So there is a continuous series Of points in which g-lines possess points of inflection between the two points in which horizontal tangents may be drawn to dp dax? g-lines must possess points of inflection on the left of the curve dy dax? side turned to the z-axis just after it has left the starting point. If 7? = 0. But there is also a continuous series of points in which the = 0, so with smaller value of z. For every q-line has its convex ae yw : ay ae it is to enter the curve — = 0 in horizontal direction and to pass at then to smaller volume, it must turn its concave side to the z-axis in that point, and so it must have previously possessed a point of inflection. Most probably the last-mentioned branch is somewhere continuously joined to the first mentioned one. If so, there is a closed d?v curve in which = 0 — and then it may be expected that this dx? g closed curve contracts with rise of 7’, and has disappeared above a certain temperature. But these and other particulars may be left to a later investigation. We have now described the shape of the g-lines, 1. in the case d dp that neither ae nor TE 2 3 ibis = 0 exists, 3. in the case that the ae ae found. dxdv de* It remains to examine the course of the q-lines when both curves dy dp de? Oped dadv is equal to O, 2. in the case that the curve == (exist. 2 2 aw eae hi d'a For the occurrence of the a = 0 it is only required that a av & be positive, as we shall always suppose, and that 7’ is below the 2 value of the temperature at which the curve pii 0 has contracted & to a single point. It may, therefore, occur with every binary system, without our having to pay attention to the choice of the components. (734-) dw dp But the occurrence of the curve ={ — }=0 is not always pos- dadv de), sible, as we already showed in the diseussion of the shape of the isobars. If we consult fig. 1 (These Proc. IX p. 630) it appears d that the curve (2) = 0 does not exist throughout the whole width a], of the diagram of isobars. With mixtures for which the course of the isobars is, as is the case in dp the left side of the figure, the line (Z) = 0 does not exist at all. das), Only with mixtures for which the course of the isobars is represented by the middle part of fig. 1 it exists and if the asymptote is found, it can occur with all kinds of volumes. Also with mixtures for which the course of the isobars is represented by the right part of fig. 1, it exists, but then only at very small volumes, and it possesses only the branch which approaches the line v = 6 asymptotically. Let us now consider a mixture such that the curve =—= () 15 Kd v 2 d wy ey da? exists; then we have still to distinguish between two cases, i. e. 1. when the two loci mentioned do not intersect, and 2. when they really present at such a temperature that also the curve d, do intersect. If they do not intersect, and the curve (2) == U lies Hij d? = 0, then the g-line, after having had its maxi- wv v on the right of ie = ae ; d mum and minimum volume, will intersect the line (Z en & v that point of intersection will have a tangent // v-axis; it will further run back to smaller volume, just as this is the case with one of the q-lines drawn in fig. 2. This may e.g. occur for mixtures cor- responding to the left region of the diagram of isobars, when this region is so wide that also the asymptote and a further part of the curve (2) = 0 is found. If with non-intersection the relative position v/)y dp dp of the two curves =0 and |—]=0 are reversed, this can da? da v probably not occur but for mixtures which correspond to a region of the diagram of isobars which has been chosen far on the right side. The course of the g-lines which then pass through 2 d the curve = 0, is represented in fig. 5. But when the two & 50 Proceedings Royal Acad. Amsterdam. Vol. IX. ( 732 ) Fig. 5: a” =") and (Z) =) de da), intersect, which then necessarily takes place in 2 points, the shape of the g-lines is much more intri- cate. Then numerator and deno- minator are equal to zero in curves dp dv de? dv j aw and (5) is not ge to be determined from this equa- dv tion. Then (5) must be deter- da le mined from : dp dv\? 9 dv + dw zE daedv? da Ac dx*dv deg ie ae In the discussion of the shape of the p-lines we came across an a” A analogous case when the curves gia and —,, av = 0 intersect ; dadv and there we found that the two points of intersection had a different character. For one point of intersection the p-line has two different ; ; yp dw da. real directions, depending on the sign of a RN: dv® dvd? this expression was negative, the loop-isobar passed through that point of intersection. In the same way, when from the above : ; dv quadratic equation for | — ; de) we write the condition on which the roots are real, we find the condition : oy dp dp î tl da® dede? \ dado negative; which may be immediately found from the condition for the loop of the loop-tsobar, as require- ment for the loop of the loop- g-line, when we interchange « and v, dadv? Fig, 6. ( 733 ) The g-line which passes through the first point where the above condition is negative, has, therefore, a real double point, and runs round the other point of intersection before passing through this double point 3 for the second time. In Fig. 6 the dotted closed curve ot = 0 has Ak dp Py been traced, and also the dotted curve | — |= — +30.) The da dvda point of intersection lying on the left, is the double point. According 3 db Buea to what was stated before, — is negative in this point, and the Ak A ee ie EE quantity 53 positive, which is also to be deduced from what was avav 5 1 : i i dp dy 4 [ mentioned previously about the sign of = — So the eri- y dudv dadv? dv terion by which the reality of the two directions of (5) is tested is da), satisfied in tbat point of intersection. In the second point of inter- dd. ME dp section ze Ì positive, and Lv ae is also positive. It is true that it avav dz" dedv* dvdx? appears in the drawing of the loop-g-line that there is no other possibility but that it runs round the second point of intersection, and 2 it appears, that just as we have mentioned in the analogous case for the shape of the p-lines (Footnote p. 626), only when the dp dy a yey does not follow from this that B — but 1 it 2 f an ; KP: wy two points of intersection coincide, so when the two curves —,=0 & =0( touch, the quantity d dw dw Bu dw an — — : vdv de? dadv? 2 ) is equal to dvda? aw da? 0. In the case that — 0 has greater dimensions, so at lower temperature, the loop-g-line will, of course, extend still much more to the right, and the higher q-lines must be strongly compressed at d, the point where the curve (3) = 0 cuts the second axis (the line v v This loop-g-line determines the course of all the other g-lines. d? Thus in fig. 6 a somewhat higher q-line passes through =O, ae in vertical direction just above the double point, rises then till it 50* ( 784 ) passes through this curve for the second time, reaches its highest dp deb sene: point, after which it meets the curve (2 = Oin vertical direction, BJ and then pursues its course downward after having been directed horizontally twice more. It must then again approach asymptotically ‘that value of a, at which it intersected the line fz dv == 0 shortly after the beginning av of its course. This line has also been drawn in fig. 6. It is evident 2 d that it may not intersect the curve v= 0. In fig. 6 it has, accord- AT ingly, remained restricted to smaller volumes than those of the curve dp da? =0. For the assumption of intersection involves that a q-line in d could meet the locus (5 dv = 0 several times. Asq= MRT EE & & such a meeting point, it follows from this that only one value of « can belong to given g. It deserves notice that in this way without 4 ; . iw any calculation we can state this thesis: “The curves rE haa 0 and AX d ; 8 f = dv = 0 can never intersect.” According to the equation of state av it would run like this: “The equations : db N° d'a db da = es MEE eae 1 dx _ de 2 (1—x) (vb) MRT VU DE can have no solution in common. Indeed, if v from the second equation is expressed in z and 7, and if this value is substituted in the first equation, we get the following quadratic equation in MRT’: d'a MRT) 1 1 1 db dx? ae 2 (1—2) += b fe da 1 (day? _ sates ia) ee A value of MRT, which must necessarily be positive to have ( 735 ) 1 db da 1 da significance, requires — — — > — —. 5 eee? ze da de~ 26 da* From the foregoing remarks it d ay d. is sufficiently clear that + must be positive to render the locus | dv U U 2 2 da a dp = 0 possible, and that — must be positive to render = 0 possi- da? da* ble. The roots of the given quadratic equation, however, are then : 5 q 6 1 db da 1 da bei 1 1 imaginary, the square of [> — 7 — 57 7, being necessarily smaller 1 db da i, ; than the square of ea and the square of this being smaller vax 1 fda ? than the product of =(z) and the factor of (MRT). ve But let us return to the description of the course of the remaining qg-lines. There is, of course, a highest g-line, which only touches the 2 d ~ locus in = 0, directed horizontally in that point of contact, and U d'r for which also de? = 0 in that point. There is also a g-line which dp ae: touches the locus ike eee in its downmost point, and which as a C lv? rule will be another than that which touches it in its highest point. The g-lines of higher degree than the higher of these two have again the simple course which we have traced in fig. 2 (p. 635) for that dp q-line which intersects the locus & = 0. Only through their con- ax) y siderable widening all of them will more or less evince the influence of the existence of the above described complication. The g-lines of lower degree than the loop-g-line have split up into two parts, one part lying on the left which shows the normal course of a q-line ee: dp ; which cuts (2) = 0; and a detached ciosed part which remains U /v enclosed within the loop. Such a elosed part runs round the second dp 2 point of intersection which & — 0 and dy = 0 have in common, daz? dt] > 2 passes in its lowest and highest point through = 0, and dx? d through (2) = 0 in the point lying most to the right and most to q av v the left. With continued decrease of the degree of q this detached ( 736 ) part contracts, and disappears as isolated point. This takes place before q has descended to negative infinite, so that q-lines of very low degree have entirely resumed the simple course which such dp 1 lines have when only the curve (2 == 0 exists: dae), Also in this general case for the course of the g-lines we can form an opinion about the loeus of the points of inflection of these ie Ae: ; d curves, so of the points in which (3) = 0. We already mentioned v EAA dp srt above that when the line (£) =— 0 exists at a certain distance from HH v it there must be points of inflection on the q-lines at larger volume. dp : é : If also the asymptote of (7) — 0 should exist, also this series ot a7 2 points of inflection of the g-lines has evidently the same asymptote. In fig. 6 this asymptote lies outside the figure, and so it is not represented — but the remaining part is represented, modified, however, in its shape by the existence of the double point. The said series of points of inflection is now sooner to be considered as consisting of two series which meet in the double point, and which have, therefore, : dp got into the immediate neighbourhood of the line (3) == there: aay So there comes a series from the left, which as it approaches the dp : : double point, draws nearer to (2) = 0, and from the double point av there goes a series to the right, which first remains within the space 9 d in which = 0 is found, and which passes through the lowest da? point of this curve, but then moves further to the side of the second dp component at larger volume than that of the curve (FZ = (The ak double point of the q-loop-line is, therefore, also double point for the locus of the points of inflection of the g-lines, and the continuation of the two branches which we mentioned above, must be found dp : above the curve a ae 0. Accordingly, we have there a right branch, de), J2 . . . a ) . . . which runs within —— —0Q, and passes through the highest point of AX this curve, and a left branch which from the double point runs to the left of the loop-q-line, and probably merges into the preceding ( 737 ) branch. If this is the case the outmost q-lines on the two sides, both that lying very low and that lying very high, have no points of inflection. THE SPINODAL CURVE AND THE PLAITPOINTS. The spinodal curve is the locus of the points in which a p- and dp dp dv dv dedv de? a q-line meet. a these points Ee ger and so RET a, dv? daedv dv? da? points of contact, we shall have to trace the p and the g lines together. As appears from fig. 1 p. 630 the shape of the p-lines is very different according as a region is chosen lying on the left side, or in the middle or on the right side; but the course of the g-lines in the different regions is in so far independent of the choice of the regions that g—. always represents the series of the possible volumes of the first component, and g4. the series of the possible volumes of the second component, and also the line of the limiting volumes. As the shape of the p-lines can be so very different we shall not be able to represent the shape of the spinodal line by a single figure. Besides the course of the p-lines depends on the dp Pw existence or non-existence of the curve ar = av av dede dp dp = x Es. 2 ). In order to judge about the existence of such = 0, and the course of the g-lines on the existence or non-existence of the curve My d P 3 ; ; ae = 0, and besides, and this holds for both, on the existence of Av wp ries , the curve = 0. Hence if for all possible cases we would illustrate arav the course of the spinodal eurve in details by figures, this examination would become too lengthy. We shall, therefore, have to restrict ourselves, and try to discuss at least the main points. Let us for this purpose choose in the first place a region from the left side of the general p-figure, and let us think the temperature so low, so below (7%),, that there are still two isolated branches for d the curve — =0 all over the width of the region. ‚U ( 738 ) In fig. 7 7 is thought higher than the temperature at which dp de below this temperature. In fig. 7 all the q-lines have the very simple course which we previously in- dicated for them, and the p-lines the well-known course, with which == 0 vanishes, and in fig. 8 (=) is positive on the liquid side p di of —= 0, and on the vapour side v of 22 0, negative between the two branches of this curve, the dv P transition of & from positive to dz}, negative taking place through infinitely large. The isobars p,, p, and p, have been indicated in the figure, in which p‚, <p, <p, Also a few g-lines, q, < g, and the points of contact of p, to g, and of p, to qg,. Also on the vapour side a point of contact of p, to q,. It is clear 1s that every g-line yields two points for the spinodal eurve, and 2°¢ that these points of contact lie outside the region in which d ee is positive. On the other hand we see that the distance from Vv Fig. 7. dp the spinodal curve to the curve ree 0 can be nowhere very large. av Only by drawing very accurately it can be made evident that on the vapour side the spinodal curve has always a somewhat larger dp volume than the vapour branch of the curve = 0. °In the tour U d i points, in which = 0 intersects the sides, indeed, the spinodal ) line coincides with this curve. Fig. 75 has been drawn to give an insight into the circumstances at the plaitpoint. At 7’ >(7;), the two branches of the curve dp =" have united at that value of z, for which 7’=(7;),. One av 5 of the p-lines, namely that of the value p=(px)z, touches in the ( 739 ) point at which the two branches have joined at a volume v = (wy), and has a point of inflection there. Two parts of q-lines have been drawn as touching the p-line. The two points of contact (1) and (2) are points of the spinodal curve, and lie again outside the curve = 0. For a higher p-line these points will come closer together. And the place where they coin- cide is the plaitpoint. As in ah a Go fo a point (1) | 7, Oa =), Te- d? d? versely in point(2,( 7) = (3): dx? ]q dx* } 5 ZEE re Fig. 7b. = ze) en (1) and (2) have coincided, and this may be considered as the criterion for the plaitpoint so that in such a point the two equations: 5) es 5 da), ° de }q (a), -@) dx* Jp dx” Ja hold. The following remark may not be superfluous. In point (2) dv dv — | is not only smaller than | — |, but even negative. In order to dx* Jp dx? Jg find the plaitpoint, the point in which 2 points of contact for the q and 25E d*v d the p-lines coincide, and so (=) and ( v Pp and dv =) have the same value, av q 2” . . . . . (3 must first reverse its sign in the point (2) with increase of & P dv the value of p for the isobar before the equality with ee) can a q be obtained. And that, at least in this case, this reversal of sign must take place with point (2) and not with point (1), appears from 20 : the positive value of 7 ) So we arrive here at the already known U q theses that in the plaitpoint the isobar surrounds the spinodal curve, and also the binodal one. and f de za AEN gn) (Owes — fr Ti == 7 aw” etc, En \ da daly i ERN 1.2.3 \de°), 1 we find for a plaitpoint : d°v dv en dz* etc. FATRA, So the p- and the q-lines meet and intersect in a plaitpoint, and this is not always changed when a point should be a double plait- point. We shall, namely, see later on that the criterion for a double plaitpoint is sometimes as follows: dv a dv dx p ~— \ dx dv d*v Eee dx p de q Let us now proceed to the discussion of the case represented by fig. 8. Here it is assumed that 7’ lies below the temperature at 3 dvy — dvg = == and d EEE — 0 vanishes, so that this locus exists, it being moreover WU which ( supposed that it intersects the curve == 0. It appears from the av drawing that for the g-lines for which maximum and minimum volume occurs, two new points of contact with the p-lines are necessarily found in the neighbourhood of the points of largest and smallest volumes at least for so far as these points lie on the liquid side of dp 0: dv So there is a group of qg-lines on which 4 points of the spinodal curve occur, and which will therefore intersect the spinodal curve in 4 points. The two new points of contact lie on either side of aw — 0, and these two new points of contact do not move far away dx from this curve, the two old points of contact not being far removed d from sip. dv If we raise the value of g, the two new points of contact draw 12a nearer to each other. Thus e.g. the g-line which touches in its ‚U dv d'n highest point, and for which G = 0 and also 2) — 0 in that q a 2 da da q ( 741 ) point has also been drawn in the figure. Also this g-line may still be touched by two different p-lines, which, however, have not been represented in the drawing. For a still higher q-line these points would coincide, and in consequence of the coincidence of two points d*v of the spinodal curve a plaitpoint would then be formed. bed always Az v 2 at . . . a v . . . . being positive, (3) , which has been negative for a long time in q the point lying on the left side, must first reverse its sign before it can coincide with the point lying on the right -— a remark analogous to that which we made for the plaitpoint that we discussed above. If on the other hand the value of g is made to descend, the point of contact lying most to the left will move further and further from Jats Ik dp ‘ 0 and nearer and nearer to the curve a7 0, till Av v the curve for g-lines of very low degree, for which as we shall presently see, the number of points of contact has again descended to two, the whole bears the character of a point of contact lying on the liquid side. But something special may be remarked about the two inner points of contact of the four found on the above q-line. When the q-line descends in degree, these points will approach each other, and they will coincide on a certain g-line. Then we have dv av again a plaitpoint. In this case neither B nor (Ga) need reverse du* J, dx” } » its sign because these quantities have always the same sign for each of the two points of contact which have not yet coincided, i.e. in this case the positive sign. But in this case, too, there is again besides contact, also intersection of the p- and q-lines. On the left of this plaitpoint the q-line lies at larger volumes, on the right on the other hand at smaller volumes than the p-line, the latter changing its course soon after again. from one going to the right into one going to the left. This plaitpoint, however, is not to be realised. With the two plaitpoints discussed above all the p-line and all the q-line, at least in the neighbourhood of that point, lie outside the spinodal curve, and so in the stable region. In this case they lie within the unstable region. Summarizing what has been said about fig. 8, we see that there is a group of gq-lines which cut the spinodal curve in four points. The outside lines of this group pass through plaitpoints. That with the highest value of g passes through the plaitpoint that is to be realised ; that with the lowest value of q passes through the plaitpoint that is not to be realised. All the g-lines lying outside this group intersect (. (os) the spinodal curve only in two points. If, however, the temperature chosen should lie above (7%), the g-lines of still higher degree than of that, passing through the vapour-liquid-plaitpoint, will no longer cut the spinodal curve. And finally one more remark on the spinodal curve, which may d? a occur in the case of fig. 8. By making the line EEn and ki dv? da” dp 3 — 0 intersect, we have a region, in which both Si and is ne- v x gative. In such a region the product of these quantities is again *) . If this should be the dadv case, it takes again place in a locus which forms a closed curve. Within this region there is then again a portion of the spinodal curve which is quite isolated from the spinodal curve considered. With regard to the p- and g-lines this implies, that there both positive, and it may become equal to ( oD and = is negative; and so that contact is not impossible. Such wv a“ p q a portion of a spinodal curve encloses then a portion of the y- surface which is concave-concave seen from below. If we consider the points lying within the spinodal curve as representing unstable equilibria, the points within this isolated portion of the spinodal curve are a fortiori unstable. The presence of such a portion of a spinodal curve not being conducive to the insight of the states which are liable to realisation, we shall devote no more attention to them. It appears from this description and from the drawing (fig. 8) that in this case the spinodal curve has a more complicated course than it would have if the curve db de a portion on the liquid side in which it is forced towards smaller volumes. There is, however, no reason to speak here of a longitudinal plait. We might speak of a more or less complicated plait here. But we shall only use the name of longitudinal plait, when we meet with a portion that is quite detached from the — 0 did not exist. It has ( 743 ) ordinary plait, which portion will then on the whole run in the direction of the v-axis. There remains an important question to be answered: “What happens to the spinodal curve and to the plaitpoints with increase of temperature ?” At the temperature somewhat higher than (7%), there exist 3 plaitpoints in the diagram. 1. The realisable one on the side of the liquid volumes. 2. The hidden plaitpoint also on the side of the liquid volumes. 8. The realisable vapour-liquid plaitpoint. Let us call them successively P,, P, and P,. Now there are two possibilities, viz. 1. that with rise of the temperature P, and P, approach each other and coincide, and the plait has resumed its simple shape before P, disappears at Y=(7%),; and 2. that with rise of 7’ the points P, and P, coincide and disappear, and also in that case the plait has resumed a simple shape. In the latter case, however, the plait- point is to be expected at very small volumes, and so also at very high pressure. Then, too, all heterogeneous equilibria have disappeared at T—(T;),. Perhaps there may be still a third possibility, viz. 2 d when the locus 7 us =— 0 would disappear at a temperature higher U than (7%),. Besides the plaitpoint P, another new plaitpoint would then make its appearance at T'=(T}), on the side of the first com- ponent. This would transform the plait into an entirely closed one, d? 5 ú = 0 vanishes, all he- and only above the temperature, at which terogeneous equilibria would have disappeared. Let us now briefly discuss these different possibilities. -We shall restrict ourselves to the description of what happens in those cases, and at least for the present leave the question unsettled on what properties of the two components it depends whether one thing or another takes place. If P, and P, coincide, the portion of the locus d? ot = 0 which we have drawn in fig. 8 for smaller volumes than & d that of = = 0, must have got entirely or almost entirely within RE d? the region where dv is negative in consequence of the rise of tem- dw perature, or the whole locus = 0 may have disappeared with wv rise of 7. . Now at P, in the previously given equation : ( 744 ) 1 dv dv Ee dv, —— dv, =a 1.2.3 | aa En EE ; da the factor of dz* is negative, but at P, this factor is positive. If the points P, and P, coincide, this factor = 0. With coincidence of these plaitpoints, called ‘heterogeneous plaitpoints by Kortewee, besides dv dv d i halt dv | d®v (3 a= |) = || S= | —— er NON ae En e& p de” Jy da* J, \da* Jq 3 es aw : 3 If P, and P, coincide, ae —0O has contracted with rise of tem- v Pw : : perature. Also eras | contracts with rise of the temperature and av is displaced as a whole, as I hope to demonstrate further. But the 3 > dw 3 contraction of 4 = 0, whose top moves to the left, happens rela- A) tively quicker, so that e.g. the top falls within the region in which 3 dx? ; : , ; ‘ 5 3 dv is negative. The existence of the point P, requires that i an 5 8 Ow wy is positive. The point P, lies on the right of aes 0 and above Se 0. L v 3 3 ‚dp ( EEN dw { If the top of at O lies within the curve me = 0, neither P, nor av Hi P, can exist any longer. Before this relative position of the two curves they have, therefore, already disappeared in consequence of their coinciding. Also in this case the coincidence of heterogeneous plaitpoints holds. At P, the factor of dx’ was positive, and af P, this factor is negative. In case of Tv dv coincidence —| — |. With de?) dz* Jo further rise of 7, however, the d? AE will have to get dv? again outside the region where a? a? is negative. The curve kk =) & dx? top of namely, cannot extend to e= 0, d? and the curve nl = Nat T=(7), v has its top at #=—= 0. We draw Fig. 9. from this the conclusion, that ( 745 ) : ; é aw with continued increase of temperature the curves —— =0 and AL iy dv? by fig. 9. The spinodal line runs round the two curves, and so in conse- : ‚dw nde quence of the presence of ~ =O it is forced to remain at an Av =0 will no longer intersect, but will assume the position indicated dp exceedingly large distance from the curve ae at = 0. The question may be raised whether the spinodal curve cannot split up into two is B d | separated parts, one part enclosing the curve es 0, the other part av 3 C passing round — = 0. The answer must then be: probably not. av Mw Mw and dv? da In the points between the two curves are indeed, posi- tive, but still small, whereas does not at all occur in the figure, dz dv and will, therefore, in general, be large. Now if the temperature at which dp Jp = 0 disappears, should lie above (7), = 0 shifts to the left, dx? dv? till it leaves the figure, and the spinodal curve is closed at «— 0 and 7'= (7%), and the new plaitpoint makes its appearance, which we mentioned above. From this moment we have a spinodal curve with two realisable plaitpoints. The graphical representation of the curvature of the p- and the q-lines is in this case very difficult, because both groups of lines have only a slight curvature. If, howe- ver, we keep to the rule, that the p- and the g-lines envelop the spinodal curve at realisable plaitpoints, we conclude that the value of da? de dv Gu . ded heek ga, 2 and 7,2) 18 positive in P,, and negative in the other plait- p q point. When these points, called homogeneous plaitpoints by KortEwse, Pv lv ; coincide, —~)=(—)=0. Above the temperature at which this dx* Jp dx? Jg takes place, the p- and the q-lines have no longer any point of contact. In 43 : : aw . consequence of the disappearance of the locus ey aie the course av of the p-lines has become chiefly from left to right, so in the direc- tion of the «z-axis. On account of the disappearance of the locus ( 746 ) Pw da? least with a volume which is somewhat above the limiting volume, they run chiefly in the direction of the v-axis. Many of the results obtained about the course of the spinodal curve, and about tbe place of the plaitpoints, at which we have arrived in the foregoing discussion by examining the way in which the p- and the g-lines may be brought into mutual contact, may be tested by the differential equation of the spinodal line. This will, of course, also be serviceable when we choose another region than that discussed as yet. From : dp dp | 0 dv? dx? dadv) dpd day dy 0 dy | aa ~} dv —() the course of the g-lines has also been simplified, and at we derive: da? dv? | dv* datde dado dede? i: at dw Be dp oer ay v? 2 dado? dx? dx*dv = di dp dn dw dy dh | Ee de” dóre "ado > or dadv da + We arrive at the shape of the factor of dT’ by considering that from: de = Tdy—pdv4+qdez follows : dy = — 4 dT — pdv + qdu dw dw d*y so that — n and so —— = t va Ge nand so — = (3 is etc. This very complicated differential equation may be reduced to a simple shape. Let us for this purpose first consider the factor of dv. By substi- tuting in it the quantity es ay ledv Py d da dv sedis for een and EE for — En dp de de), dp dv? dv? this factor becomes : Bw (dw (dv \’ yp (dv dw Nma ene = eal ane dv? | dv*\ dz p dadv* \ dx p dae*dv d From p = — ~ we derive: Vv ( 747 ) dw (d 7 a ( + pith — dv da p dudv Py (d?v Py (dv \? By (dv dey 2 2 ie 73 EE di - 2 BE aE J 2), dv* \ da? Jy dv* \da/y dv*da \ de}, da*dv from which appears that we can write the factor of dv in the form of: Bared de? dx? : We might proceed in a similar way with regard to the: factor of dx, but we can immediately find the shape of this factor by substi- tuting the quantity w for v and q for p in the factor of dv. We find then: dy (3 ~\ de? dv? . As long as we keep 7 constant, and this is necessary for the course of a spinodal curve, the differential equation, therefore, may be written : d? 2 d? d? 2 13, zel spake ibs Wai fe de dr == dv? dx? Jp dx? dv* }q 4 ; ax da \* (d?v : By taking into account that | — |= — {| — | | —— |, we obtain dv°)g dv ]q\ da” q after some reductions which do not call for any explanation, the simple equation : dv dv pa dv de? q dx ae dx al P dz? and As a first result we derive from this equation the thesis, that dv dv Dey AL dy — and | — must have the same sign, if | — | and | — de) spin dit)» — gq da? Jy dx* Jo have the same sign and vice versa. Thus on the vapour side in fig. 7 dv d?v ; dv : — | and | — | have always reversed sign, and | — being dx*}, dax* } 4 da j dv negative, | — 7 ) is negative on the vapour branch of the spinodal Gk / spin p curve. Reversely the curvatures of the p and q-lines have the same 5 DOEN, do dv rr sign on the liquid side, and | — ie = positive. If, however, dx bin dx p= 51 Proceedings Royal Acad. Amsterdam. Vol. IX. ( 748 ) d*v , — | should have been indeed negative there, as was accidentall dz’ 5 q represented in point 2 of the spinodal curve, the spinodal curve runs towards smaller volumes with increasing value of z. So if there occur points with maximum or minimum volume on the spinodal curve, dv dv Te. V2: — | ==0 in those points. If on the other hand {| — is infinitely de? Jg de J spin large, which occurs in the case under consideration when the spinodal 2 dv curve is closed on the right side for 7 >> (7), then oe) must be wv p —0, and so the p-line must have a point of inflection in such a point, to which we had moreover already concluded before in another way. A great number of other results may be derived from this differential equation of the spinodal line. We shall, however, only call attention to what follows. In a plaitpoint (=) =(215 d spin p=q dx Py For a plaitpoint it follows from this that (= N= 2 any da” Pp dx? ’ ; dv wee : If for a point of a spinodal curve ea is indefinite, both spin Q wv 2 2 G ;) and Ge) must be equal to 0. This takes place in two cases: du? }, du? ]q 1. in a ease discussed above when the whole of the spinodal line is reduced to one single point. 2. when a spinodal line splits up into two branches, as is the case for mixtures for which also 7’, minimum is found. In the former case the disappearing point has the properties of an isolated point, in the second case of a double point. In the differential equation of the spinodal line the factor of dT may be written: 1 dT dp 5 Ty Py Ty a Ei de? Jr dv’ dedv Jp dadv in dvu* Jar dx? \ and by putting e — ® for Ty it may be reduced to: 1 (d'W de dap de dw de en da* dv? or to 1 dw {de /dv\* Ge (dv ad? ne ra do? =() = A dadv (5) zn as = Pd Dad) ld’w The factor by which — — Tan is to be multiplied, occurs for the ( 749 ) first time in formula (4) Verslag K. A. v. W. Mei 1895, and at the close of that communication I have written this factor in the form : ; 2 Id Lda) | aay v day 2a dx a? from which appears that in any case when a,a, >a,,?, this factor is negative. Here, too, I shall assume this factor to be always negative, but I may give a fuller discussion later on. In consequence of these reductions the differential equation of the spinodal curve may be written as follows: Py (dv je Le dv ph ay, dv? dx), Ay tages Ay ae From this equation follows inter alia this rule concerning the displacement of the spinodal curve with increase of 7, that on the 2 d*v Me side where (=) is positive, the value of v with constant value of 2a Vv x, increases, and the reverse. So the two branches of a spinodal curve approach each other with increase of the temperature. But I shall not enter into a discussion of the further particulars which might occur when this formula is applied. By elimination of dv I shall only derive the differential equation of the spinodal line when we think it given by a relation between p, « and 7. We find then: av d, av dv dr dv dp=(* (ee), da a deter: (3) da? de v, 7 dx? pT da gr \ dp /, da pa for a plaitpoint the factor of dz disappears, and we find back the equation (4). Verslag K. A. v. W. Mei 1895, for the plaitpoint curve. At constant temperature we tind for the spinodal curve : / d'v\, EN 2 justen de spin dx v a . P de (To be continued). 51 ( 750 ) Physics. — “The shape of the empuric isotherm for the condensation of a binary mixture’. By Prof. J. D. van DER WaAats. Let us imagine a molecular quantity of a binary mixture with a mass equal to m,(1—#) + m,r, at given temperature in a volume, so that part of it is in the liquid state, and the remaining part in the vapour state. Let us put the fraction which is found in the vapour state equal to y. The point that indicates the state of that mixture, lies then on a nodal curve which rests on the binodal curve. Let the end of the nodal line which rests on the liquid branch be denoted by the index 1, and the other end by the index 2. Let us represent the molecular volume of the end 1 by v,, and the molecular volume of the other end by v,, then when v represents the volume of the quantity which is in heterogeneous equilibrium : v=v (ly) +0 y the constant quantity z being represented by : e=, (1—y) + 4, y- From this we find: dv = (v,—v,) dy + (l—y) dv, + y dr, and 0 = («,—~2,) dy + (1—y) dz, + y de By elimination of dy we obtain the equation: y) de, + y de) — (l—y) dv, — y dv. v — 1 = WH dv dv Now in general dv = @ de + (=) dp. Let us apply this equa- a Jp ap /y tion for the points 1 and 2 of the binodal curve, and let us take the course going from v, to v, + dv, and from v, to v, + dv, on the surface for the homogeneous phases. Then: dv dv, = dp, ; (Ge) ee . ae e ) ae - 5 dv, oo ee dix, Jp Te ee dv dv The quantities (3) and (=) must then be taken along an isobar. Pp dz, Jp dx, If we substitute the values of dv, and dv, in the equation for dv, it becomes : and ee d = do = (= y) [EE (A) | NC a) Er ev, da, Pp dp ta | yeas lv 1 Oel Oa, da, p dp hom GE) res v,—?, dv, ee) G is (= | i ed ——— — —" — mn dp het 9 Lt, dz, Pp - bin dp hom | (7 =) dp hom Bs Now the factor of (F je = : (F). and we find: dp Join %,—2, da? ]»,T bin EI (Ls) Dn RS =) — q ) | + dp er J an? pT bin dp hom dE da. \* HD ap (1) ; de? jee A dp an i hom If we consider the Me of the condensation, and so y= 1, the above equation becomes: Se ted G) dp het I mn dp bin dp hom or in which we must put v,=v and z, ==. It appears from this dv dv equation, that never (5) = —( ) , and that there must dj ) / het dp hom dv d therefore be a leap in the value of aug of ee at the begin- Pp Vv ning of the condensation, unless there should be cases in which as dae ; : ort: ‘ == — | is equal to 0. The only case in which this is so, is de?) \ Ap] vin yy i dp dx in the critical point of contact. There | — | =o and so | — |= 0 aL? bin dp bin But then there is properly speaking no longer condensation, and the empiric isotherm has disappeared. We might think of a plaitpoint, aag AN dp because | —— = 0 in it, but on the other hand == and di pT da bin de zes VS da? — | =o there. If the limiting value of Is or of dp bin de? v dp bin is sought, we find by differentiating numerator and deno- minator twice with respect to z: a ae aS da? pln 3 da? pT dec’ p,T DE WG). da } vin de ] yin \ dz? _] bin da* ] yin iy Aba aS aS d'5 In a plaitpoint, besides | — | also | — == @ but | 2 da nk de ps da gd will have a value differing from 0, and so there is a leap in the dv value of — — in a plaitpoint too. dp dv Sn dv As —{— must always be positive, also Sin) will always dp hom het dv dp d be positive and larger than 5 a) or (— 2) >(- ee) : dp hom dv J hom dv J het At the beginning of the condensation the empiric curve will ascend less steeply with decrease of volume than that for homogeneous phases. , dv dp There are cases in which (5) = Ge OF (- el == 0) ae dp) het dv / het 2 „2 d's ees tS on the sides, so for-.«—=0O and z==1. Then &. is infinite, and AL an p MRT is represented by the principal term ( i 2. if on the binodal da dp 4 curve — is infinite or — — 0; this takes place for those mixtures dp dx which behave as a simple substance. If in equation (1) we put y=0O and v, =v and 2, = « we could derive the same conclusions for the end of the condensation. eh : dp dp MO, The relation between — | — and — | — at the beginning dv J het dv hom and at the end of the condensation, could be immediately derived . by applying the equation : 3 dv : dv ; y —— EEK 7 ) ee Q v < dp) le dx p both for the surface of the homogeneous phases and for that of the : : 2 v heterogeneous phases. If we then take into consideration that (as) Av p 2 . Ua a > on the heterogeneous surface is equal to — —, we find: U, 1 ( 753 ) V,—?, dv dv dv — de + | — Jdp =| — | de + | — | dp, B, dp / het dx)» dp ) hom and from this the former relation. dv\ . : From the form for (- — } in general, so not only at the begin- het dp ning or at the end of the condensation, we see that the empiric isotherm can have an element in which it has an horizontal direction only when a nodal curve is intersected, at one or the other of CN TONG : : whose ends & is infinitely large. But as neither the sides nor ap / bin the nodal curve which runs parallel to the v-axis can be intersected, it would follow that the empiric isotherm can never run horizontally in one of its elements. There ave, however, cases which form exceptions to this rule. First of all if we widen the idea empiric isotherm, and understand by it the section of a surface // v-axis with the derived surface of the y-surface, also in the case of a complex plait. Then there are also nodal curves to points in which the binodal curve passes through the spinodal, and where therefore du ( 2 oo. But as such equilibria are hidden equilibria, they cannot P) bin be realised in spite of this. Instead of this we have rectilinear inter- - section of the surface // v-axis with the three phase triangle, and in ; dv\ . A LADE this part ra 3s of course, again infinitely large. But secondly, yp het and this is a case which might, indeed, be realised, the binodal dx curve has a point in which ( =o, when this point is a plait- \ dp bin point which with increasing or decreasing temperature will become a hidden plaitpoint. This is a limiting form of the first mentioned ease, in which the three phase triangle was intersected. Then the three phase triangle has contracted to a single line, and the above mentioned straight line has contracted to a single point. Then there is, of course, a point of inflection of the empiric isotherm in that point. With larger volumes it is curved negatively, with smaller volumes positively. ( 754 ) Physics. — ‘“‘Jsotherms of di-atomic gases and their bindry mixtures. VI. Lsotherms of hydrogen between —104° C. and — 217° C.” By Prof. H. KAMERLINGH Onnes and C. BRAAK. Comm. N°. 97¢ from the Physical Laboratory at Leiden. (Communicated in the meeting of December 29th 1906). $ 1. Introduction. The investigation treated in this Communication forms part of the investigation on the equation of state of bydrogen, which has been in progress at Leiden for many years. *) With that part of our measurements ?) which we now deem fit for publication, we have more directly carried on the work that H. H. Francis HyYNDMAN had already done with one of us (K. O.) before 1904, so that, though all the observations, one for this, another for that reason, but always for the purpose of reaching the desired accuracy (which, we may add, was increased in the course of the investigation) have been repeated, an important share of the final success of the measurements is due to the said investigator. The results obtained by us furnish data for applying the correction of the readings of the hydrogen thermometer to the absolute scale experimentally (see the following communication), and for determining the deviation between the net of isotherms of hydrogen and that of the mean reduced equation of state (see Comm. N°. 71, June 1901 and Comm. N°. 74, Arch. Néerl. 1901) *). The points determined in 1) In Comm. N’. 69 (March 1901), where the apparatus have been described which were used in this investigation, the Communications referring to this subject, have been mentioned. Since then the isotherm for 20°C. to 60 atms. was given in Comm. N°. 70 (May and June 1901) with the accuracy of which the open standard- manometer (Comm. N'. 44 Oct. 1898) and the closed standard manometers (Comm. N°. 50 June 1899) admit, which investigation is carried on in Comm. NO. 78 (March 1902) for the isotherms of hydrogen at 20° C. and 0°C., which have been determined with the apparatus that have also been used for this investigation. The suitability of these apparatus for accurate determinations of isotherms has been shown in Comm. N'. 78, and is confirmed by this Communication for low temperatures. Several communications e.g. N°. 83, 84, 942 and 94/, further 85 and 95, finally Nes. 89, 93 and 95 are more or less in connection with this investigation, the great importance of which, if accurately carried out, is demon- strated in Suppl. N°. 9. 2) We soon hope to publish the results of measurements at higher pressures and lower temperatures, and also those of supplementary determinations at lower pressures. 3) Definitive values for the virial coefficients B and C (§ 12 contains only provi- sional values) from which the difference with those according to the reduced equation may follow, are given in the following communication. ( 755 ) the net of isotherms are only few in number, but these few points have been determined with particular care, so that, so to say, they form normal places in the examined region of the equation of state, with which without preliminary adjustment we may set about the calculation of individual virial coefficients. Characteristic of them is that every group of such normal places obtained by deter- minations with the piezometer and manometer (see Comm. N°. 69 and 78) lies really on the same isotherm (that of about — 104°, — 136°, — 183°, —195°, —205°, -—213° and —217°), and that on these same isotherms every time a point has been obtained at small density by a determination with the hydrogen thermometer (see Comm. N°. 95¢ Oct. 1906). The great difficulty ') of this investigation lies in obtaining the required constancy and stability of the low temperatures. Accordingly the arrangement of reliable cryostats was made a separate subject of investigation at Leiden. (cf. Comm. Nes. 83 and 94). This investigation comprises three series of piezometer-determina- tions at densities respectively about 70, 160, and 300 times the normal.’) Several of the observations mentioned here lie in the neighbourhood of the curve of the minima of pv. They enable us to determine the shape of this curve pretty accurately (see § 13). We confine ourselves in this communication to our observations themselves. A discussion of them, also in connection with the results of other observers, will be given in a following communication. § 2. Survey of the apparatus used. a. On Pl. I in fig. 1 we find a schematic *) representation of the system of the apparatus for measurements and auxiliary arrangements, excepted those which serve for keeping constant the temperature in the cryostat. The compression apparatus A is the same as that mentioned in Comm. N°. 84 (March ’03). For the meaning of the system of tubes, cocks and other parts we may refer to Comm. N°. 69 and N°. 84. The same figures have been used, except that in this communication ¢c, is used for the cock which admits the 1) Witkowski, whose important investigation on the expansion of hydrogen (Bulletin de l'Académie des Sciences de Cracovie 1905) had already appeared before the experiments mentioned in this communication had been completed, already mentions this as an explanation for the fact, that he has dropped the direct determination of isotherms at temperatures lower than — 147°. 2) The limits are chiefly given by the pressure under which the gas stands ; they are about 20 and 60 atms. 3) The individual apparatus are represented on the same scale, the connections schematically, ( 756 ) compressed air, and c, for the cock which shuts off the level glass. Of the compression tube, provided with the system of cocks, mercury- reservoir and level-glass belonging to it a front-view is given in fig. 3 of Pl. I. The piezometer with the connections g, and g, has been represented in detail in fig. 2 of Pl. I. The arrangement of the cryostat B which has served for the deter- minations mentioned in this communication, is described in Comm. N°. 944, For the description of the apparatus serving for keeping the tem- perature in the eryostat constant, we may also refer to this last communication. Fig. 4 of Pl. I may also serve for elucidating this description for the special case that our piezometer is placed in the cryostat. The pressure is conveyed (see fig. 1, Pl. I) from the compression- tube to the manometer along c¢,,, C,, Co Cig» Co, And ¢,,. By closing and opening c,, the differential-manometer /*) may be shunted in and out. By means of the cocks c, and c,, it may be shut off from the rest of the pressure-conduit, when great differences in pressure are brought about, or are to be feared. *)*). The apparatus are placed in two rooms as has been indicated in the figure by a dotted line. By closing one of the cocks c,, and c,, the two parts may be treated as independent systems. This was done when the manometer was compared with the open manometer connected at c,,. The manometer C is the same as served for the investigations of Comm. N°. 78. The reservoir D serves, if necessary, for eliminating the injurious influence of small leaks, for which purpose it is placed in ice. At c¢,, it can be coupled to the system. In the experiments of this communication there was no need to use it *). The pressure is exerted by compressed air, which enters through c¢, and c¢,, along the drying tubes /and G; and is regulated by blowing off along c,,. The cocks, Er Ein Gp Cee ANG lave analagous meaning to ORO Cis 0, VAN es 1) This manometer, which was formerly used with the open standard-mano- meter, (see CG fig. 1 Comm. N°. 44) and had now been removed to the piezo- meter, was of great use for finding leaks. 2) A couple of mercury-receptacles, which served for receiving the mercury that might overflow, have not been represented in the figure. 3) The system which we have so far described and which belongs to the piezo- meter, is placed in one of the rooms of the laboratory, situated in the immediate neighbourhood of the cryogen department. The remaining apparatus which chiefly belong to the manometer are erected in the room with the standard-manometer. 4) The adjustment of trays of oil for the different couplings rendered the search for leakages so easy, that an injurious leak needed never to remain, ( 757 ) All couplings of the conducting tubes in which air is to be kept at constant pressure, have been placed (cf. the plate to Comm. N°. 94%) in trays filled with oil, according to what has been said in Comm. N°. 94°. b. With regard to the means for keeping a constant temperature in the eryostat, the system of pumps and auxiliary arrangements for the regulation of the temperature, belonging to the circulation of oxygen, has been represented in fig. 4 of Pl. I. For a description we refer to Comm. N°. 94. Some particulars about the ethylene circulation used for the deter- minations of Series I, are to be found in Comm. N°. 94/ XIII § 1. § 3. The manometer. The pressure measurements were performed by means of the closed auxiliary manometer described in Comm. N°. 78°. As a comparison of this manometer made in 1904 with the standard manometer A IV (of Comm. N°. 78 $17), yielded an unsatisfactory result, and led us to expect that the auxiliary manometer was no longer reliable, it was compared at four points with the open standard manometer, to which the improvements mentioned in Comm. N°. 94° were applied *). The results of this comparison have been combined in the subjoined table. Column C like column C of table XVII of Comm. N°. 78° repre- sents the reading of the pressure determined with the open mano- meter (Comm. N°. 44). Every value is the mean of two observations. Column # gives the pressure read by means of our closed auxiliary manometer. Each of the values has been obtained as a mean from three observations. In the calculation the calibration derived in Comm. N°. 78° has been used. In column G the difference of the columns F and G is represented, column H contains this same difference expressed in the numbers of column Cas unity. The pressure given by the auxiliary manometer appears to be too high for all pressures observed. It was obvious to ascribe this to a too high value assumed for the normal volume ’). If we take the mean of the values in column MZ, we find 0.00087. If we diminish the normal-volume and so also the pressures by this part of the original amount, the differences represented in 1) In the investigations with this manometer of Comm. N’. 70 the total absence of leaks was rare; here, however, it was easily brought about. Also the improved coupling of the steel capillaries to the glass-capillaries of the open manometer by platinizing proved satisfactory. (See Comm. N°. 94%), 2) In connection with this diminution of the normal volume see also Comm. NO. 95e, § 11. ( 758 ) column K remain between the indications of the auxiliary manometer and the open standard manometer. These differences, considerably smaller than those in table XVII of Comm. N°. 78’, remain within the limits of accuracy fixed for this investigation, and justify us to estimate the mean error in the pressure measurements at + annie In the following the pressures have been calculated with this new value of the normal volume. TABLE I. Manometer. ia aa | e | F | G H | K 94.947 94 64 + 0.017 0.00070 = 000017 36.290 ie + 0.043 0.00120 + 0.00033 47.960 48.004 + 0.044 0.00092 + 0.00005 oe | ost | toom 60.029 60.061 + 0.039 0.00065 — 0.00023 § 4. The piezometers and auxiliary apparatus. The piezometer used in the first series for the observations at a density 70 and the temperatures —104°, —136°, —183°, —195°, was of about the same dimensions as that used for the observations of Comm. N°. 78. In the subjoined table, just as in the corresponding table II of that communication, the dimensions are given to facilitate a survey of the amount and the influence of the many corrections. TABLE II. Data H,, Series I. U,=6.4140 cms. Py=— a0 ; U5 0: 0520 => Mak er ea Vn V4=6.0174 » U, has been determined from a calibration table Ve = 516017 cm?. D'r 05722 py “neren: 1) The values of @ given here have been determined for the ordinary temperature and those for lower temperatures have been put equal to them. We hope soon to determine (3 also for lower temperatures. ( 759 ) The stem 6, (see Pl. IT Comm. N°. 69), on which the volume U, is read, was 30 em. long in order to enable us to determine every time three points on the isotherm which did not lie too near each other *). In others of our piezometers it was taken still longer. For the series II and HI a piezometer of larger dimensions was used. The necessity of the use of a larger gas-volume for determinations at densities higher than 120 times the normal has already been men- tioned in § 19 of Comm. N°. 84. The volumenometer described there was not used,’ but just as in Series J the normal-volume was deter- mined in the piezometer itself. As in the preceding table the dimensions of the piezometer are given here. TABLE III. Data H,, Series III. UF 75) 1583 emt £.=3.7 A0 Oi 00382 7 B AOR Wi Us 1.407" 5 VJ=10.9645 » U,= (see preceding table) V, = 2063.30 cmö. woes > 1.617 > per em. In Series II the piezometer-reservoir had a volume of 10.343 eM?, but for the rest it had the same dimensions as in Table III. To detect any escaping of gas during the measurements at high pressure in consequence of leakages at the connections g, and g, (see Pl. IT Comm. N°. 69), cylindric glass oil-trays were placed round these couplings (see Pl. I fig. 2) which enabled us to discover immediately even the slightest leakage; everywhere the oil-trays rendered excellent services, but here they were of the greatest importance for obtaining reliable results *). 1) They may serve, inter alia, to give us information about the curvature and the inclination of the isotherms at the middle point. 2?) Once the oil-trays near the couplings gj and gg rendered good services, when before the determinations of Series III gas escaped in consequence of the nut gy being imperfectly screwed on. From a determination of the normal-volume made immediately afterwards, it proved to have changed so much that the previous determinations had to be rejected. ( 760 ) §5. Lhe hydrogen. The filling was accomplished for Series HI with all the improve- ments described in Comm. N°. 94e § 2. For the first series the purification by means of cooling in liquid air was not yet applied, in the second series it was, but without application of high pressure. §6. The temperatures. The temperatures ¢, and ¢, respectively of the divided stem 5, and the steel capillary g, (see Comm. N°. 69 PI. II) were determined in the same way as in Comm. N°. 78 $ 13. In series I three ther- mometers were placed along the steel capillary, and one at the part of the glass capillary f, that remained outside the cryostat. The refrigerating action of the cryostat proving to be very slight even in the immediate neighbourhood, only three thermometers were used in the following two series, two at the ends and one in the middle of the steel capillary. The influence of an error of 41°C. in the temperature of the capillary (comp. Comm. N°. 78 § 13) is only sie at —200°. 000 of the total compressed volume at — 100° land 4000 For the temperature of the glass capillary we assumed here that indicated by the thermometer at the end of the steel capillary. This simplification is the more admissible as the temperature in the cryostat is lower, and hence the volumes outside it contain less gas. The temperature of the glass capillary in the cryostat has been determined in the same way as was followed in the investigations with the hydrogen thermometer mentioned in Comm. N°. 95’. As the arrangement of the cryostat was the same in the two cases, and the measuring-apparatus placed in it had almost the same form, there was no objection to start from the previously found data for the determination of the temperature of the capillary. (see Comm. N°. 952 $4). This method gives sufficient accuracy, as, reasoning in a similar way to that followed in the said communication, we arrive at the result, that an error of 50° in the temperature of the part of the stem that is taken into consideration still gives a negligible error in the final result, viz. less than 5000, The temperature ¢, of the piezometer-reservoir was determined by means of the resistance-thermometer, which (ef. Comm. N°. 95°) had beforehand been compared with the hydrogen-thermometer. They differ little from those at which the calibration of the resistance- thermometer took place. Hence the reductions are simple and may be effected with great accuracy. The temperatures were not calculated directly from the resistance ( 761 ) formula of comm. N°. 95 § 6, but they were based on the separate readings of the hydrogen-thermometer, because the latter must also serve as points of the isotherms. From the above mentioned formula i hide determined, and by the aid of this factor the reduction was effected. The difference of temperature for which the reduction was made amounting to less than 0.°3, this method of calculation is perfectly sufficient ; only for the temperature of — 1357.71, where the difference amounts to 4°, another correction of 0.°01 was required. In the subjoined table *) the method of calculation has been repre- sented for one determination of the temperature. The first column con= tains the observed resistance W, in the following column JV, represents the resistance at which the resistance-thermometer has been compared with the hydrogen-thermometer (of comm. N°. 95° Table I § 6), and T represents the corresponding reading of the hydrogen-thermometer. From the value and W— W, follows now the temperature-correct- aw ion At which is to be added to 77, in order to give ¢, on the hydrogen- thermometer-scale. TABLE IV. Temperature of the bath during observation N°. 7 of Series III. 5 Te ae | Ww | We | ij | = | At | t, | 17.295 | 17.290 | — 912°832 1.750 | + 0.009 | — 949-89 | § 7. The measurements. At the beginning and the end of every series the normal-volume was determined in the way described in Comm. No. 78 § 12, only with this difference that in the series mentioned here every deter- mination of the normal-volume was supplemented with a reading in the U-tube 6, and one of the barometer. In this way two determi- nations were generally made before and after every series. The values ; : spe lt found before and after every series, differ nowhere more than 3000" The tables V and VI are analogous to the tables VII and VIII of Comm. N°. 78. In the former the results are represented referring to 1) The difference of the numbers in this table with those of the Dutch text is due to an improved calculation. The influence of this improvement enters also in some numbers of the last part of this communication. ( 762 ) the determination of the normal-volume for series III, the latter com- prises the three series. Se TABLE V. Normal volume #,, Series III. N°, Volume. Pressure. PvA | Mean. | Mean. | Difference. 4 | 4955.78 75.546 4915.56 OR 1945.58 » 1955.82 715.545 1945.69 4945.73 =— (15 9 4961 .39 75.342 1945.87 4945.87 + 0.14 | 40 [1962.04 715.327 1946.13 1) + 0.40] 44 [1962.31 15-317 1946.14 + 0.41] TABLE VI. Normal volumes A, Determinations, | Before. | After. | Mean. Difference. nd N°. 4 — HT 9 544.74 544.82 544.78 UD Series I 10 + 0.02 441 + 0.07 4 — 0.60 2 | 1945.73 1946.10 — 0.46 3 0-03 44 1946 .16 ?) + 0.05 Series II ( 42 — ().48 94 | 1946.10 1946 .64 + 0.39 99 + 0.58 95 + 0.47 Series III 4945.58 4945 .87 1945.73 see Table V 1) The two last determinations have been left out of account, though they show but slight deviations, because on account of variations of temperature in the room a certain cause could be assigned for the apparent rise of the normal-volume. 2) A determination of the normal volume was made in this series both before and after, and also between the determinations at high pressure. The value given here is the mean from these three determinations. ( 763 ) As far as the pressure permitted, three points were chosen in every series on every isotherm for the determinations of the isotherms ascending with about equal differences of density, which offers advantages for the calculation of the virial coefficients (see § 12). The readings were adjusted at these points by bringing the mercury at the bottom, in the middle and at the top of the divided stem 5, By way of control the two points in the middle and at the bottom of the stem were for some determinations determined once more with decreasing pressure. For every determination we waited till both the temperature and the pressure were constant, and we could assume that the equilibrium of temperature and pressure had been established. This will be the case when the meniscus in the divided stem is moving up and down within the same narrow limits. The stability of the tem- perature was ensured by a good regulation, and that of the pressure was easily obtained and preserved by paying attention to the oil- trays mentioned in § 2, which immediately betrayed the slightest leakage *). When the above mentioned constant state had set in, some readings of the piezometer and the manometer were alternately made. If they agreed, we proceeded to the next point. With regard to the regulation of the temperature the measure- ments took place under the same circumstances as the investigations with the hydrogen thermometer described in Comm, N°. 95¢ (Oct. 1906). Besides the resistance-thermometer for regulation and deter- mination of the temperature the thermo-element was also used here by way of control for the determination of the temperature. The indications of the resistance-thermometer, however, proving more reliable than those of the thermo-element, only those of the latter apparatus were used for the calculations. All possible care was always taken that the temperatures at which determinations were made, lay as close as possible to those which have been used for the calibration of the resistance-thermometer to render the corrections small, and the accuracy of the determination of the temperature as great as possible. The regulation of the pressure took place according to the indication of the metal manometer M of Pl. I, fig. 1. If we passed but slowly from one pressure to the other the thermal process in the reservoir which attended it, was so slight, that the regulation and the measure- ) Formerly this often required a long and sometimes fruitless search (cf. e.g. comm. NY, 70 p. 8). 52 Proceedings Royal Acad. Amsterdam. Vol. IX. ( 764 ) ment of the temperature did not experience any perceptible disturbance, on account of which the stability of the bath was the more ensured throughout a whole determination of an isotherm. § 8. Calculation of the observations. The calculation is made in the same way as described in Comm. 78 § 8. For the calculation of the variability of the volume of the piezo- meter-reservoir U, at low temperatures we started from the qua- dratie formula for & found in Comm. N°.95/$ 1, so that 4, 93.43. 4 10 and. hi AOT 10-56. No correction for glass- expansion was B for the volume of the glass capillary U, nor for that of the steel capillary U,. For the reduction of the gasvolume in the glass capillary U, to 0° we proceeded as follows: the volume was divided into 5 parts Usa, U», Us, U2a and U, Usa represents the part in the liquid bath increased by 2 em. of the capillary above the liquid, where the temperature may still be put equal to that of the bath. Us, Us, and Usa form together the remaining part of the capillary in the cryostat above the bath, Us, + Us, corresponds to the volume w', of § 5 of Comm. N°. 95e and Uog to w",. Use is the volume of the capillary that is outside the cryostat. For the reduction of the volumes U», Us, and Usa we started from the same determinations of the temperature as in Comm. N°. 95°. As on account of the greater density at lower temperatures the mean tem- perature found cannot directly be used for the reduction, each of the above mentioned volumes is divided into 3 parts, the temperature of each of these parts is derived, and from this the mean temperature t is determined according to the formula == in which T represents Op the absolute temperature. The coefficients of expansion, which are required for the reduction, were determined by means of the general development into series of Comm. N°. 71 with a slight modifi- cation of the coefficients mentioned there. The results obtained in this way do not give an appreciable difference with those which were found when the reductions are made with the approximate results for the determinations of the isotherms obtained in this way. With regard to the corrections of the temperature of the volumes U, and U, we proceeded in a way similar to that of Comm. N°. 78. For that of the reservoir at low temperature a somewhat different way ( 765 ) was followed. As practically the temperature for every individual determination of the temperature might be considered as constant (see $ 1), a number of parts of isotherms could be immediately obtained for every series separately. As they at the same time refer to about the same densities, an accurate value may be derived for E pva) dt temperatures in the different series not differing more than 0.°2, the results could be reduced to one and the same standard-temperature in this way without the slightest difficulty. As standard-temperatures were assumed the temperatures —103°.57 and —135°.71 of series I, —182°.81 and --195°.27 of series Il and —204°.70, —-212°.82 and —217°.41 of series HI. In the subjoined table the values of 1 | from the graphical representation of pv4 on —. The VA VA 1 4 PVA" pPvar TEN are given, which served for the reduction of the Mt 0 » A remaining determinations of the isotherms to these standard tem- peratures f,, which relate to the hydrogen-thermometer at constant volume and 1100 mM. pressure at 0°. fre TABLE VIL. H, (a) * Series IL. | Sj Oek, —195°.27 9042 23 | 219° 98 Density. Gy Cp Cp | 489°. 81 }_195°.97 9049.83 150 0.004336 =| 0.004406 0.004501 160 0.004390 0.004458 0.004540 170 0.004440 0.004513 0.004588 184 0.004508 0.004599 0.004667 | From these mean coefficients values for E ed could be derived dt hry for the different points of the respective isotherms, which for the isotherm of —212°.82 have been given in Table XI of § 10. § 9. Survey of a determination. As an instance of determinations of isotherms at low temperatures we give here one of the measurements from the 3 series at a density 326 times the normal, in oxygen boiling under strongly reduced pressure. 52* ( 766 ) TABLE VIII. H, Series HI -N° 7. Determination in oxygen at about — 213° C. Time 3.10—3.20 | A B | C | D | E | it | G H | K Piëzometer top | 5.8) 56.496 | — 2139/209.76| 19°.5 rim | 4.8] 56.360 | — 188° 49°.4| 21.6 division n° 29 | 7.0] 56.864 | — 98° 19°. 4 Ae Manometer 5.0 |} 93.97| 19.98 | 20.04 Piëzometer top | 5.9/ 56.493 | 20° .76} 49° .4 rim | 4.8} 56.360 2176 19°.3 Manometer ; 5.0 || 93.97] 20.00 90.05 Float 1e The columns of the table agree in the main with those of table IX of Comm. N° 78% A and B have the same meaning, C denotes the temperature ¢, of the piezometer-reservoir in the bath, and the temperatures fz, toe and #7 in the order of the parts Uo, Ur, and Us. (see § 8) of the glass stem in the cryostat above the bath. The temperatures given in this column are to be considered as constant throughout the determination, and have, therefore, been mentioned only once. D gives the temperature ¢, of the waterbath round the stem b,, EH the temperatures 4, 7," and f,” of the thermometers placed along the steel capillary. The temperature of the part Us, that projects above the cryostat is put equal to f,. The columns F, G and H have the same meaning as in the above mentioned table. In column K the indication of the float in the cryostat has been given. All lengths are in ems. These readings are corrected in the same way as was followed for table X of Comm. N°. 78°. These corrected values are given in the following table. The two readings of the piezometer have after- wards been united to a mean after reduction. ( 767 ) TABLE IX. H,, Series III. Determination at about —213° C. Corrected and recalculated data. - A | 2 c}p]z|r|e H LE Manometer mean | 115.5//93.97/19.935 Temperatures f,, ¢,, ts —213°|20°.70}18°.6 top —188° (2e — 980 t2d — 11° 28.634/0.135 82.3 Piëzometer 28.631/0.132 Surface of the liquid | 1.9 The columns from A to H (H included), have the same meaning as those of table X of Comm. N°. 78’. K denotes the position of the liquid level above the boundary of the piezometer-reservoir and the glass stem, derived from the indications of the float. From this the following table is obtained, which gives the cor- rections of volume and pressure required for the calculation of pv4 . It corresponds to table XI of Comm. N°. 78. The volumes of the parts of the glass capillary with their correc- tions have been separately given. Moreover the corrections w', and h have been added, the former is a consequence of the packings being pressed down at g, and g,, the latter accounts for the weight of the compressed air in the connecting tube between the manometer and the piezometer. The vertical distance of the levels of the mercury is about 0.5 meters. Instead of the mean coefficient of expansion hk, the double term &,-+4,¢ has been assumed (ef. § 8) for the computation of the correction w,. Here we must point out that as standard temperature f, for the reduction of the parts into which the glass capillary is divided, + 20° has been assumed, so that the differences ¢,—1#, are not very small here. The method of interpolation applied in Comm. N°. 78 for small values of ¢,—/, will be used as soon as we have tables for d.,, at our disposal. The values have been directly determined here by calculation. ( 768 ) | TABLE X, H,. Series III. Corrections and final result. Dar 4.4904 em’. pm = 61.004 atm. Wi = 0.0003 » a 0.434 » wy = 0.0014 » h =—0.004 » ui = — 0.0104 » UB == 0.0018 » ws = — 0.0005 em}. Uva = 0.0018 » ua = 0.0059» Uy = 0.0020 » UG = 0.0049 » Uo = 0.0025 » UI = 0.0024 » Usd = 0.0129 » ud —= 0.0015 » Use = 0.0196 » Ute —= 0.0001» wi = — 0.0239 » wi = 0.0012 » (pva)t, = 0.18863 t, ==. — 2120.82 (pv4)__ 94190 82 = 0.18863 for p = 61.434 atms. The total value of the correction of the stem appears to be very small, so that we might apply the law of Gay—Lussac down to— 217° without introducing appreciable errors. § 10. Values of pva. The values of pv, obtained in this way for the different determi- nations have been represented for the isotherm of —212°.82 in the following table. The values in the last columm refer to the reduc- tion to the standard-temperature ¢, (ef. the conclusion of $ 8). The values of this table have been obtained, as appears from table IX, by calculation with the mean values of the separate read- ings. The deviations in these separate readings which may be due both to oscillations of the pressure and the temperature and to errors in the readings themselves, amount nowhere to more than 2000: The result found for about the same point at the beginning and at the end of one determination of an isotherm are in very good accordance, as moreover is to be seen by comparing observation ( 769 ) TABLE XI. H,. Results for the isotherm of —212°.82. id Cl | TEEN Ne: ty P i PVA da ee BED 17 — 194708 30.591 0.19406 157.64 0.00458 18 35.426 0.19134 185.15 0.00473 Series II 19 33.071 0.19264 171.68 0.00465 20 30.554 0.19405 157.46 0.00458 — JA2°.89 51.632 0.48767 playa (2 Series III | 61.434 0.18863 325.68 N°. 17 and N°. 20 of table XI. The results are reduced to the ‚| deva) same standard-temperature by means of the values of | A given in the last column. In table XII the results obtained in this way are also given for * the remaining isotherms. Those belonging to series | are less certain and will be repeated. The results obtained at the beginning and at the end of a determination of an isotherm at about the same density have been united to a mean. For every temperature we have added to the results of the deter- minations of isotherms those of the readings of the hydrogen-ther- mometer to which the former are in direct relation. The numbers do not agree with those of the preceding table, because some determinations have been united to a mean, for which reason they are indicated by ( ). The points of the hydrogen-thermometer have been obtained in the following way. From ScHaLKwiJK’s determinations of isotherms follows for 20° C. pva = 1.07258 + 0.000667 d4 + 0.00000099 d4*. If we suppose the mean pressure-coefficient from 0° to 20° not to deviate appreciably from the value 0.0036627 between 0° and 100°, which is permissible on account of the insignificant deviations of the indications of the hydrogen-thermometer of constant volume from the absolute scale, it follows from this that: (pv) 0°, 1100 mm. = 1.000275. The value given in Comm. N°. 60 having been taken for the pressure-coefficient of hydrogen for the calculation of the hydrogen ( 770 ) TABLE XII, H.. Values for (pra), No | ts | p Ì pra da —————— H, therm. (1) | — 103°.57 | 0.8°6 | 0.62082 1.444 (2) 32.985 | 0.63467 | 541.971° Series I (3 39.659 | 0.63765 | 62.193 (4) 49.897 | 0.64274 | 77.632 H, therm. (1) | — 135°.74 0.727 | 0.50307 1.445 | (2) 28.592 | 0.51064 | 55.994 Series I (3) 33.437 | 0.51258 | 65 231 H, therm. (1) | — 1827.8! 0.479 | 0.23054 1.448 (2) 46.572 | 0.32700 [142.42 Series II (3) 55.293 | 0.32822 168.46 H, therm. (1) | — 195°.27 | 0.443 | 0 29486 1.449 (2) 40.599 | 0.27367 [148.35 Series II { (2) 45.484 | 0 27337 [166.36 (4) 49.998 | 0.27343 [182.85 Hs therm. (1) | — 2049 70 | 0 363 | 0.95031 1.449 | (2) 35.487 23189 [153 03 Series IL { (3) | 38.640 23097 {167.30 (4) 42.438 „23010 [184.43 „23009 [269 10 „22056 1.450 0 0 0 Series III (5) 61.917 | 0 0 (2) | 30.689 | 0.19480 |157.64 0 0 0 0 0 H, therm. (1) | — 212°.82 | 0 390 Series II { (3) 33.200 19339 |171.68 (4) 35.566 19210 [185.15 (5) 51.632 18167 |275.42 Series III (6) 61.434 18863 [325 68 H, therm. (1) | — 79.41 0.295 20375 1.450 (2) 46.419 | 0.16381 |283.814 Series III | (3) 52 898 | 0.16336 [323 80 |» 58.971 | 0.16424 [359.04 me thermometer-temperatures, the value of pv4 at ¢, now follows from the formula (pva)t, = (pva), (1 + 0.0036627 ¢,). § 11. Probable error of a determination. The mean error in the calibration of the large volume of the piezo- ; 1 ; : meter may be estimated at + 2000: As to the volume into which the gas is compressed during the measurements, the greater density of the gas in the reservoir at low temperatures may be allowed 1 for by reckoning only with 5 of the amount of the volumes at the temperature of the room. The errors in these volumes being predo- minant with respect to those in the volume of the piezometer reser- voir, the mean error for measurements below —180° with piezo- 1 meters of 5 c.M* may be put equal to + Enon of the compressed volume in accordance with the degree of accuracy as was calculated in Comm. N°. 69, where for measurements at the ordinary temperature 1 the mean error is estimated at + fon for piezometers of 5 c.M*. | pee 5 ror will be + ——. For 100° the mean error will be SE The mean error of the determinations of the normal volume is an that of the measurements of the pressure may also be j. estimated at + ——. 3000 In the determination of the temperature there is no appreciable error. The observations made for one point show that the mean error due to variations of temperature and faulty readings of the position i 1 of the mercury in the stem, may be put smaller than + 5000" The mean error of the determination of temperature in the stem 1 remains below + ——. 6000 The mean error caused by all these sources of errors together x amounts to = 1500 for piezometers of 5 e.M.? and not very low 1 : temperature, to + EE for larger piezometers and very low tempe- ( 772 ) ure. The different points on one and the same isotherm must show smaller discrepancies inter se than corresponds with the said mean error. The mean error namely, for a determination, apart from the errors ‘in the determination of the normal volume and the 1 calibration of the large volume is + fror IT to =e rs, All this does not apply to the isotherms of —103°.57 and —135°.71. These belonging to series I are the earliest determina- tions and for different reasons less accurate than the later ones. § 12. Provisional individual virial coefficients. If the temperatures had not been given as readings on the hydrogen- thermometer of constant volume at 1100 mm. pressure, but on the absolute scale, the coefficients A4, B4 etc. calculated from the equation Bg C4 Da BA RA RE en ee ooh VA VA VA Vv A vA with the values of pv4 from table XII, could be immediately com- pared with those derived in Comm. N°. 71 *). However, this is not the case, because the latter relate to the absolute scale of tem- perature. From the outset it has been our purpose to derive the correction of the hydrogen scale on the absolute scale experimentally — from our measurements themselves. This might be attained by first neglecting the correction, and by caleulating provisional values A'4, Ba, C'a ete. for each of the isotherms, which serve then for finding provisional corrections for the hydrogen-thermometer ; after this the calculation is repeated with the corrected temperatures, etc, till further repetition would not bring about any change. A similar treatment has been applied for the determination of the corrections of the readings of the hydrogen-thermometer to the absolute scale, where we purposed to draw through the observations for every isotherm a curve, which does not only correspond as closely as possible to the observations, but also to the general equation of state. In this § 1) We must call attention to the fact that in the calculations of Comm. N°. 71 we began by taking 273°.04 by first approximation for the absolute zero-point ; we should find the correction to this from the results of the calculations of iso- therms, and then proceed to a second approximation. We have still retained 273°.04 in VI. 1 Suppl. N°.8 and in VI. 2 Comm. N° 92. Since then, however, a set of coefficients VII. 1, which will be published in the following communi- cation, have been calculated with the further approximation for the absolute tem- perature, viz. the more accurate value 273°.09, and corrections have, moreover, been applied in critical quantities etc. ( 773.) the method of least squares has been applied directly to the indivi- dual isotherms, in order to obtain a formula which represents the observations as accurately as possible. The number of points on each isotherm not being large enough for all six coefficients to be determined at once, definite values were assumed for the last three values. F4 was put = 0, and values were calculated for D4 and 4 from the sets of coefficients VILA *), which was chosen instead of V of Comm. N°. 71. This assumption means, that a definite course was prescribed for the isotherms at higher densities, which corresponds as closely as possible to the law of the corresponding states. The results of these calculations are laid down in the subjoined table. D4 and £4 are the values assumed for the calculation according to the above. TABLE XIII. H,. Provisional virial coefficients. ts Al 103. Bla 40°.Cla | 40°.D4 | 40'8. Brg — 4039.57 0.62048 0.94971 | 0.558% | 0.9113 | -— 0.648 — 135°.71 | [0.50303 0.03234 | 41.7974 | 0.7028 | — 0.408] — 182°.81 0.33063 | — 0.08384 | 0.4021 | 0.3809 | — 0.088 _195°.97 | 0.98503 | — 0.13051 | 0.3565 | 0.9802 | — 0.016 — 2049.70 | 0.25058 | — 0.18030 | 0.3710 | 0.2166 | 0.034 — 9199.82 | 0.29090 | — 0.29433 | 0.3668 | 0.4544 0.066 | area | 0.2040 | —0.95013 | 0.375 | 0.1499 0.082 | It appears from the table, that the coefficients of the same column vary regularly with the temperature, except for — 135°.71, for which we may account by taking into consideration that the two piezo- meter-determinations which had to be used for the calculation, lie so close together, that a slight difference in their relative situation already produces a large difference in B'4 and C'4. By the aid of the coefficients the values of pv4 were determined anew according to formula (1). The divergencies for every isotherm between the assumed values of pv4, Wi and the R,; calculated with A'4, B'4 and C'4 (pva=1 for 0’ and 760 mm.), where z indicates the number of that observation in table XII, have been represented in the subjoined table. | ) For the calculation of Di and #4 the uncorrected reading of the hydrogen thermometer was used (see preceding note). (774) TABLE XIV #H,. Deviation from formula (1). | 10% (Wi—Foi) in °/, of Roi ts ca raises (aj ek faAlins i=6 —4103°.57| —1 | +7 | —9 | +3 0.001 /}0.011|0.015/0.005 —135°.71 —182°.81 —195°.27/} +4 | +1 | —4 | +2 0_00410.C04,0.014/0.007 —2049.70| —1 | +9 0 —9 | +1 0.004/0.036| 0.000) 0.036 |0.004 —212°.82} —2 | +5 | +6 | —2 | —17} +10)0.007|0.022|0.027|0.009|0.077|0.045 —217°.44| 0 0 —3 | +2 0.001 0.000)}0.014/0.010 The isotherm of — 212.°82 is best adapted to give an idea of the accuracy of the mutual agreement on account of its larger number of points. The agreement proves very satisfactory. The upper limit 1 of the mean error may be put at +. 2000 § 13. Minima of pv. By means of the coefficients of table XIII the following minima of the pv-curves were derived from the data of table XII. TABLE XV. H,. Minima of pva. ts PvA da Pp WB, — 182°.81 0.32630 102.24 “3.36 — 0.08 — 195°.27 0.27338 174.45 47.69 + 0.50 — 204°.70 0.22935 227 AT 52.10 — 0.75 — 212°.82 0.18780 285.55 53.63 + 0.26 — 170.41 0.16335 315.72 51.57 + 0.08 By means of the method of least squares the coefficients of a parabola p=P,+ P, (pea) + Ps (pea) have been calculated from these data *). They are: 1) It is to be remarked that the less certain isotherms of — 104°-and — 136° are not used in this deduction, H KAMERLINGH ONNES gases and their binary _| between — 104° C. an | | H KAMERLINGH ONNES and C. BRAAK. Isotherms of di-atomic gases and their binary mixtures, VI. Isotherms of hydrogen between — 104° C, and 21% C. Plate I. = se e B En il 2 = Wy ( a6? De Fig. 2. al Fig. 4. Proceedings Royal Acad. Amsterdam. Vol. IX. “XI [OA “wepiorswy ‘peoy [vfoy ssuIpee001q + 4 cl 09 Blerk Olne LNE O5 R oi o | | ez EE B | | | al ine Ee et 0D | | | te eo | 5 ik cae É S lan iz | ae . : t A WOO | | #9 are ==] | i | | | i = < | | ; SS [za0e==¥ | ed - | | Sf = eal | | or sGi-=] L Cee ees let ohne etal) Sa ee a 3 | x | sjees) : | a B: EE 5 = i | | | | | | | | | L | = E (Eels iL : | Er d = = | =) ad TI 931d ‘) cLIG@— PUR ‘Q oFOT— Woemyoq uosorpky JO SWIOTZOST “TA ‘soanyxim Axeurg 1104} pue Sesvs ormoge-ip Jo sWAe4}OSE 'HVVHH O PU SUNNO HONIIHENVA H ( 775 ) P, = — 2.623 En 552.610 Paste 1954006: The differences W—R, between the given values of p and those calculated with these coefficients have been represented in the last column of tabie XV. They amount to little more than 4 atmosphere. The results given in the table have been reproduced in a diagram on Pl. II"); the curve traced there is the calculated parabola. It follows further from the values of the coefficients, that the parabola cuts the ordinate p — 0 in two points, where pv 4 is respec- tively 0.00480 and 0.40307, from which follows with the formula *) (pva)r = 0.99939 {1 + 0.0036618 (7 — 278°.09)} for the corresponding temperatures measured on the absolute scale, T,=1°.3 T = 110°.2. The top of the parabola lies at a pressure of 53.73 atms. the value of pv4 is here 0.20394, from which follows, in connection with the value of (“ae dt 0.0053, for the absolute temperature of the isotherm which passes through the top that ) determined from the isotherms, viz. p = 53.73 T = 68°.5. *) Physics. — “On the measurement of very low temperatures. XIV. Reduction of the readings of the hydrogen thermometer of constant volume to the absolute scale.” By Prof. H. KAMEriincu Onnes and C. Braax. Communication N°. 97% from the Physical Laboratory at Leiden. (Communicated in the meeting of Jan. 26, 1907). § 1. Introduction. As it is till now difficult to obtain pure helium, and very easy to obtain pure hydrogen (cf. Comm. N°. 94f, June 1906), the scale of the normal hydrogen thermometer (that with constant volume under a pressure of 1000 m.M. of mercury at 0°) is for the 1) The temperatures have been given in absolute degrees below zero. The temperatures noted down on the plate undergo slight alterations on account of a more accurate calculation of the corrections to the absolute scale. They become —103°.54, —135°.67, —182°.75, —195°.20, —204°.62, —212°.73 and —217°.32. 2) This value of A 40 has been calculated from Scuatkwisk’s determinations of isotherms (cf. the conclusion of § 10). 3) Im this the corrections to the absolute scale have been taken into account. (776 ) present, just as when it (1896) was first mentioned as the basis of the measurement of low temperatures at Leiden in the first com- munication (N°. 27) on this subject, still the most suitable temperature- scale to determine low temperatures down to —259° unequivocally with numerical values, which come nearer to the absolute scale than those on any other scale. It is therefore of great importance to know the corrections with which we pass from the normal hydrogen- scale to the absolute one. As is known they may be calculated for a certain range of tem- peratures, when the equation of state for this region of temperature has been determined at about normal density. Up to now we had to be satisfied for that calculation for the hydrogen thermometer below 0° with equations of state of hydrogen obtained in a theore- tical way. BerrHeLor ') derives them by means of the law of the corresponding states from experimentally determined data of other substances in the same region of reduced temperature. CALLENDAR °) modifies vaN DER Waars’ equation of state so as to render it adapted to represent the results of the experiments of JouLE—Ketvin for air and nitrogen as well as those for hydrogen between 0° and 100°, and supposes that a same form of equation holds also for hydrogen outside this region. Chiefly this comes to the same thing as the application of the Jaw of the corresponding states, albeit to a limited group of substances. Though such theoretic corrections as have been given by BeERTHELOT and CALLENDAR are a welcome expedient to help us in default of other data’), yet an experimental determination of these corrections remains necessary. We have obtained them in this research by using the isotherms of hydrogen between —104° C. and —217° C. given in Comm. N°. 978, 1) Sur les thermométres à gaz, Travaux et Mémoires du Bureau International, T. XIII. 2) Phil. Mag. [6] 5, 1908. 8) Wrostewski’s determinations of isotherms at the boiling point of ethylene and oxygen are not accurate enough for this purpose. In the results found for the last temperature this is immediately apparent from the irregular situation of the points on the isotherm. The values obtained at the boiling-point of ethylene give more harmonious results. And yet a correction on the absolute scale would follow from them whick has the wrong sign, viz. — 0°.07. At the temperature of liquid air Travers has determined the difference of the hydrogen thermometer of constant volume and constant pressure, from which we may also derive the corrections to the absolute scale for these temperatures. It is obvious that this derivation cannot be very trustworthy. Further it is now possible (see § 1 of Comm. N°. 974) to derive data on the expansion of hydrogen at low temperatures from the determinations of Wirkowsk1; they will be discussed in a following communication. (779 For the calculation of these corrections at a definite temperature we might start from the individual virial coefficients in the development into series of the equation of state (ef. Comm. N°. 71, 1901), which we have derived in § 12 of Communication N°. 97¢. The results obtained in this way show really a regular course‘), in spite of the small number of points on the isotherms. However, we wished first to adjust the results of the separate isotherms by general formulae of temperature. Both in this case and in general it is very difficult to succeed in this by application of one of the equations of state drawn up in a finite form. Very suitable for such a purpose is the general development inte series (or more strictly speaking, development into a polynomial), which has already been mentioned frequently. We chose for this the form VII. 1 (ef. the footnote to § 12 of Comm. N°. 972). The adjustment takes place by calculating for every isotherm modifications in B and C, AB and AC, which we call individual AB and AC, with an approximate value of the correction to the absolute scale, by then representing the values of AC by a general formula of the temperature, and by computing new values for AB by successive approximation in such a way that the value for the correction on the absolute scale corresponds to the assumed value of 7. Finally also the values of AB were represented by a general formula of the temperature. If we put the new values of B and C’ obtained by the aid of these corrections, which special values we denote by VII. H,. 1 in the polynomial of state, then this represents at the same time the determinations of isotherms of Comm. N°. 70 at 20° very satisfactorily, and those of Comm. N°, 78 at 0° and 20° by approximation. By means of these general expressions the reductions on the absolute scale have been carried out. If B and C are known there is another way to derive the absolute temperature from the observations with the hydrogen thermometer, than by applying the corrections which lead from the hydrogen scale to the absolute temperature scale. In the calculation of the temperature from the observations we may namely take at once into account, that the gas in the thermometer does not follow the law of Boyrr-CuarLes, but that pressure and volume are connected in the way, as is indicated by the development into series with the corrected values of B and C. The formula which may serve for this purpose, is given in § 5. 5 Only the isotherm of — 135°.71 gives a devialing result. (See the conclusion of § 12 of the preceding communication). ( 778 ) § 2. Reduction of the readings of the hydrogen thermometer of constant volume to the absolute scale. If v is the volume of the gas in the thermometer, expressed in the theoretical normal volume, p the pressure in atmospheres, 7'the absolute temperature, the equation of state for the thermometer gas may be written in the form: pov = Ar (1 in a v v (2) B'r C'r = Further we put: ¢ the temperature on the scale of the hydrogen thermometer of constant volume and T— To ¢. = 0. ¢ is determined by zee (bek (pv), Ep where a, represents the mean pressure-coefficient between 0° and 100° for the thermometer with the specific volume v. This is given b (P),00 — (PV) Ny —$<$<$— 100 (pv), If we represent the correction on the absolute scale by: Ato it, we may write for this: T 100 B\00— Tl’, Ti00© 1co— ea T B'7— Ee B', RGS Gi eee: 100 » 100 v2 in v nn (3) Di a PooBhoo—ToB'o , Tio i00—T0'o 100 » 100 v2 In agreement with what may be derived from the mean equation of state VII. 1, it appears from our determinations, that the influence of C’r is very slight, and down to — 217° does not amount to more than 0°.0003, so that it has not to be taken into account. Therefore in what follows will be put C’7=0, as is also done by BrrtHELot but without proof. For the absolute zero point the value 273°.09 *) is assumed, from 1) From Amaaat’s experiments with the development into series of Comm N°. 71 (cf. the note to. § 12 of Comm. N°. 972) 1.26 X 10—5 was found for the difference between the pressure-coefficients of nitrogen at 1000 mm. pressure and 0 mm. pres- sure, from which follows with Cxappuis’ pressure-coefficient for 1000 mM., i. e. 0.0036744 the value 0.0036618 for the limiting value at 0 mM. pressure, corresponding to the absolute zero point — 273°.09. In the same way hydrogen gives for the difference of the pressure-coefficients at 1090 mM. and O mM. 2.1 X 10—6, which with the pressure-coefficient 0.0036629 given in Comm. NO. 60 (see XV) gives (779 ) which follows A7—=0.0036618 7’, Tec, =273°.09 and T'iooec.—=373°.09. For the reduction of the data given in Comm. N°. 97¢ to the ; VA theoretical normal volume the value — = 0.99939 was taken, borrowed v from the determinations of isotherms of Comm. N°. 70 (Scnarkwijk). The values of B’, and ZB, have been derived from the same determinations of isotherms') by the aid of the pressure-coefficient 0.0036629 (see XV at the end of this Communication), neglecting the correction to the absolute scale for 20°. These values are: ?) B', = 0.000607 —-B’,,, — 0.000664 The values of B'p were found from the VII. H,.1 already more fully discussed in § 1, which gives in a reduced form *) 1 1 10° D= + 178.247 ¢ — 462.956 — 706.416 — + 384.2458 = — 4.2530, t whereas VII. 1 gives: 1 1 1 LOD 157.9500 ¢ — 305.7713 == Se asa — 97.5686 2a 4.2580 a From this the values of B'r have been calculated for the standard temperatures of the isotherms. The subjoined table contains in the first column these standard temperatures ¢; measured on the scale of our hydrogen thermometer,‘) the limiting value 0.0036608. The same value as was found above from nitrogen, was derived by Berrueror (loc. cit.) from Crarpurs’ results for nitrogen and those for hydrogen obtained with a thermometer-reservoir of hard glass. In the same paper he derives the value 275°.08 for the absolute zero-point for the case that also the less concordant results found by Cuappurs for hydrogen with a platinum- thermometer are taken into account. Afterwards (see Zeitschrift für Elektrochemie N°’. 34, 1904) the first mentioned value 273°.09 is again found by taking the mean of the above values for nitrogen and hydrogen, and those which may be derived by means of the experiments of Kervin and Joure. 1) Compare the conclusion of § 10 of Comm. NO. 97a, 2) The values found by is are resp. 0.000579 and 0.000606. Those of WirkowskI are 0.000616 and 0.000688. Those derived in Comm. NO. 71 from the observations of AMAGAT are 0.000669 and 0.000774. 3) According to Dewar, pA=15 atms. and 7K =29° are used for the caleula- tion, which also served for the derivation of VII. 1. Further have been put 440 = 0.99939 and 4 4= A40 (1 +0.0036618 2). 4) The slight differences with the value of table XII of Comm. N°, 97a are due to a correction (see XV) in consequence of the application of the improved pressure-coefficient 0.0036629 and the influence of the dead space on the deter- minations of the temperature, which will be more fully discussed in the last part of this communication. 53 Proceedings Royal Acad, Amsterdam. Vol. IX. ( 780 ) in the second column the same temperatures measured on the absolute scale. The two following columns contain the corresponding values of the special 4’7 and of the corrections to the absolute scale A ¢, calculated according to formula (2) for a hydrogen-thermometer of constant volume with 1090 mm. zero-point-pressure. The last column gives the corrections for the normal hydrogen-thermometer. The values for — 103°.56 and — 135°.70 are less certain than the others (compare § 10 and § 11 of the preceding communication). TABLE XVI. H/,. Corrections to the absolute scale. | Ss | t | 4 | B' AO | At | s T | — = ! | | | — 103°.56 | — 103°.54 | + 0.2892 | 0°.0214 | 0°.0196 | — 135°.70 | — 135°.67 | + 0.2368 0°.0316 | 0°.0290 | — 182°.80 | — 182°.75 | — 0.2327 0°.0530 =| 0°.0486 — 195°.96 | — 195°.20 | — 0.4734 | 0°.0614 0° 0564 — 2049.69 | — 204°.62 | — 0.7244 | 0°.0683 | 09.067 | | — 242°. 812) — 212°.73 | — 1.0142 | 0° 0752 | 0°.0690 | | — 2179.40 | — 2179.32 | — 1.2167 | 0°.0796 | | | With very close approximation the results of the last column may be represented by the formula: t benk ak og EAS Re where : a = — 0.0143507 b — + 0.0066906 e = + 0.0049175 d= + 0.0027197 The greatest deviation is three units of the last decimal. The formula gives the value Af==0, both for ¢—=-+ 100° and for t— 0°, while At—-+ 0°.14 would follow from it for ¢=-— 273°. § 3. Accuracy of the corrections. The influences which may cause errors in the corrections, are of two kinds. 1. Errors in the values of B'r. 2. Errors in the data which have been used in the further derivation. 1) The difference with Comm. N°, 97a remaining after the correction of the preceding note is the consequence of an improvement applied in the calculation. (781 ) The latter may be reduced to the error in B', and the difference of the pressure-coefficients used for the density =O and that at 0° and 1090 mM. If for the mean error in B', we compare the values of B’, which may be derived from the data of Comm. Nes. 70 and 78 and from those of CHappuis, a mean error of + 0.000034 (about agreeing with the error per cent derived for the pv in § 11 of Comm. N°. 974) follows from their deviations inter se, which corresponds with a mean error of + 0°.008 at — 100° and of + 0°.003 at — 200° for At. We may further assume that the mean error in the pressure- coefficients 0.0036618 and 0.0036629 amounts to one unit of the last decimal for the first and to two units for the second, which corresponds with a mean error in At of + 07.003 and + 0°.006 at — 100° and of + 0°.005 and + 0°.011 at — 200°. If we further put the mean error in B'r equal to that of B, a mean error in Af corresponds to this of + 0°.006 at — 100° and of + 0°.002 at — 200°. The total mean error in consequence of all these mean errors together will amount to £0°.012 for —100° and £0°.013 for —200°. § 4. Comparison of the results with those which have been theore- tical'y derived. Table XVII contains the corrections concerning the normal hydrogen TABLE XVII. H/,. Corrections to the absolute scale. At | At : Apr according ‘ | Boia A A values nj Se | Callendar | Berthelot | values | — 103°.56 | 0°.0196 | 0°.0017 || — 10° | 0°.00021 0°.0015 | — 1357.77} 0°.0290 | 0°.0032 || — 20° | 0° „00048 | | 0°.0034 | — 182°.80 | 0°.0486 02-0082 || — 50° | 0°.00164 | 0°.°082 — 195°.26 | 0°.0561 | 09.0108 || — 100° | 0°.0054 | 0°.008 | 0°.0187 — 204°.69 | 0°.0627 | 0°.0136 |] — 150° | 0°.0132 | | 0°.0337 — 212°.81 | 0°.0690 | 0°.0168 || — 200° | 0°.0344 0°.06 | 0°.0593 — 2179.40 | 0°.0730 | 0°.0192 | — 240° | | 0°.18 | 240° | 0°.0470 || — 250° | 0°.1005 | sad 090925 | 53* é Jean) thermometer. Besides the above mentioned values of At, which were directly found from the observation it contains the corrections deter- mined according to the serial formula VII. 1 and those calculated by Catrmxpar and Berrue.or. Moreover in the last column the corrections, which may be calculated from the experimental values adjusted with VII. H, according to formula (4) are given for a comparison. Besides the corrections derived from this investigation for the zero- point-pressure of 1000 m.M., also the values found by BrrtHELoT and CALLENDAR are represented on the plate. The three curves have been indicated by I, If and III in the above mentioned order. Also II and III refer to a zero-point-pressure of 1000 m.M. The values derived by CALLENDAR and Bertue.ot by means of the law of the corresponding states appear to deviate systematically from the experimental ones. With regard to the corrections according to VIL. 1., in the derivation of which formula agreement in the region of the equation of state (between 0° and —217° for hydrogen) treated here, was not aimed at, we may observe that a modification is required for VII. 1 to give as good an agreement as possible also in this region. In the first place this agreement would require that for the calculation of VII. 1 those values were assumed for the critical quantities of H, which follow from the data of Comm. N°. 974. They are p;=—= 15 atms. and Tj, = 48°. This value of 7; would considerably increase the corrections given in table XVII according for Vibe. § 5. Formula to derwe the temperature directly from the obser- vations with the gas thermometer of constant volume. We suppose that the correction for the difference in pressure at the mercury meniscus and the thermometer-reservoir in consequence of the weight of the thermometer-gas is applied to Hy, and that it is so small that it may be neglected for the small volumes. The fundamental formula for the reduction is *): By CP pv=Ar (: = ESS EE ) Vv Vv which may also be written in the form : ) (p) po = Ar (1 +B? p+ cP r) eh ds EA We start from this latter formula. The equation for the gas-ther- mometer (ef. formula (1) of § 5 of Comm. N°. 95°) becomes now : 1) Here v is expressed in the theoretical normal volume and hence AT= = 1 +0.0036618 5. We call the value for 0° C., at which 6=0, A%. lt is 1. ( 788 5 Vi(i+4, t+, t°) + B, Hu, u Hr : oe ae - | A (1+ BY) H* iy tough mm vf rw Uu, | Us 4 u, | A „(1 i! H) A, Bik HE) A, (B BY B) E te ee Butt mt a" u | ~ | A Bp Hy Cn) | ABP) | 0) This formula holds also for the carbonic acid thermometer up to the number of decimals given by Cnappuis. In XV we shall further discuss the deviation of the formula used by Crapputs. With a sufficient degree of approximation the formula for the determination of the temperature down to 0.001 with a hydrogen thermometer of 1100 m.M. zero point pressure and a dead space u = may be written in the simpler form: jz | Voth tthe) + btm ty! a A,,(1+B” H,,) 140.003661 | Tee zE Te ate ats 4 Uu, Ie | 1-4+0.00366 t,""" 1+40.00366¢, ' 1+4-0.00366¢, | _ V - ! all wr ate ot Bt ud u, u, +u, nee u | (7) A, (HBW H,) 1+0.00366 15 First an approximate value may be assumed for BY. With the approximate value of the temperature found in this way a better value of BY may be determined, and the correction term for the expansion of glass calculated. Thus we find Ar, from which the value of @ follows through Ar— Af, 0.0036618 — XV. Influence of the deviation from the law of Boyrn—Craruns on the temperature, measured with the scale of the gas thermo- meter of constant volume according to the observations with this apparatus. § 1. When the formulae are drawn for the calculation of the temperature on the scale of the gas thermometer of constant volume the variation of pressure of the gas both in the thermometer-reservoir ( 784 ) and in the dead space has as vet (see e.g. Cuapputs) been generally entered into the calculation. as if it took place at perfectly constant density. The error committed in this way, is so slight for the permanent gases for small values of the dead space, that it manifests itself only in the last of the decimals given by Cuappuis. For CHapputs’ carbonic acid thermometer, however, it attains an appreciable value (the influence extends here to the last decimal but one), so that it was of importance to examine in how far it is permissible to neglect it. This appears when Cuapputs’ formula is more closely compared with formula (6) of XIV. The density not being constant, either in the thermometer-reservoir nor in the dead space, on account of the fact that e.g. at low tem- peratures gas passes from the dead space to the reservoir, and pv as well as the pressure-coefficient varies with the density, four approximations are applied in this treatment (two for reservoir and two for dead space), all giving an error in the same direction. (Adsorption is left out of account). The errors caused by these approximations, are of the same order of magnitude for the reservoir and the dead space, the first applying to a large volume and a small difference of density, the second to a small volume and a large difference of density. The correction which is to be applied to the determination of temperature on account of these errors, only amounts to — 0°.001 at — 100° for a hydrogen-thermometer with 1000 mm. zero-point-pressure and a dead space of 0.01 W,, to somewhat less for lower temperatures, and so it may be neglected below 0°. Formula (6) differs from the preceding formula by one correc- tion more, which is independent of the size of the dead space, and which is the result of the variation of density in the reservoir caused by the expansion of the glass. This error is of no importance for the determination of the temperature by the hydrogen-thermometer, but may exercise an appreciable influence in some cases. (cf. § 3). The approximations mentioned have also an influence on the deter- mination of the mean pressure-coeffieient. The discussion, perfectly analogous to that for the influence on the determination of the tem- perature, gives + 0.00000019 as correction for our thermometer, which remains below the limit of accuracy given in Comm. N°. 60. Hence the value 0.0036627 derived in Comm. N°. 60 for hydrogen at 1090 mm. changes into the corrected value 0.0036629. § 2. We may pass from the temperatures derived in the way Prof. H. KAMERLINGH ONNES and C. BRAAK. On the measurement of very low temperatures. XIV. Reduction of the readings of the hydro- gen thermometer of constant volume to the absolute scale. À 038: DN AEN Ee ea ce RA EN = OG [vik 4 | | EA 50° ~300° #50 ° ~200° 250° t Proceedings Royal Acad Amsterdam. Vol. IX. Ke pet gs Ea Pea. be = 7 « Le. Gn am Pe == ie mi me — Sail as wei bd 7 ir: “1 - De _ a ( 785 ) mentioned in Comm. N°. 95° to those on the normal hydrogen-ther- mometer by availing ourselves of the subjoined table, in which the corrections required for this have been given. These corrections give an account of the variation in the assumed pressure-coefficient and (with regard to the number of decimals given) of the influence of the dead space. TABLE XVIII. Corrections for the temperatures calculated according to Comm. NO. 95e to those on the normal hydrogen scale. Rh as Wik eae | ae | 50° + 0°.003 — 200° | + 09.016 — 1009 + 09.006 — 220° | + 09.019 | — 1500 | + 0°.010 — 250° | + 09.020 | By means of the fifth column of table XVI the corrections to the absolute scale are found. Thus the tables XVI and XVIII enable us to reduce the temperatures calculated according to Comm. N°. 95e and used in Comm. Ne 957, 95° and 954 both to the normal hydrogen scale and to the absolute scale. The temperatures f, occurring in Comm. N°. 97%, already corrected in the first column of table XVI for the application of the corrected pressure-coefficient 0.0036629 and the influence of the dead space, are adjusted to the absolute scale by the corrections in the fourth column of table XVI. $ 3. The values found by Crarruis and Travers for the pressure- coefficient of hydrogen (ef. the footnote to § 7 of Comm. N°. 95°) are corrected to 0.00366266 and 0.00366297 (number of decimals the same as given by thein). For the pression-coefficient of carbonic acid found by Caappuis the correction is more considerable and amounts (because the dead space 1s small here, the correction on account of the variation of density caused by expansion of the glass is here about of the same value as that on account of the variation of density by the dead space) to — 0.25 « 10-6, so that the value found by Curareurs ') 0.00372624 is corrected to 0.00372599. 1) Nouvelles Etudes, Travaux et Mémoires du Bureau International. T. XLII, p. 48, ( 786 ) Physics. — “Contributions to the knowledge of the w-surface of van per Waats. XV. The case that one component is a gas without cohesion with molecules that have extension. Limited miscibility of two gases.’ By Prof. H. KaMERLINGH Onnes and Dr. W. H. Krrsom. Supplement N°. 15 to the communications from the Physical Laboratory at Leiden. (Communicated in the meeting of Februari 23, 1907). § 1. Introduction. In the Proceedings of Dec. ’06, p. 502. (Comm. N°. 965) it was mentioned that the investigation of the w-surface of binary mixtures in which the molecules of one compo- nent have extension but do not exert any attraction, would be taken in hand as a simpler case for a comparison with what the observations yield concerning mixtures of He, whose molecules are almost without cohesion. Before long we hope to give a fuller discussion of such a w-surface '). In the meantime some results have already been obtained in this investigation, which we shall give here. Thus it has appeared, that at suitable temperatures, at least if the suppositions concerning the applicability of vaN per Waars’ equation of state with a and 4 not depending on v and 7’ for constant w, mentioned in § 2 hold for these mixtures,*) two different phases may be in equilibrium which must be both considered as gasphases. Then the two substances which are the components of these mixtures, are not miscible in all proportions even in the gas state. And if certain conditions are fulfilled this may continue to be the case when the one component is not perfectly without cohesion, but possesses still some degree of cohesion, which, however, must be very slight. From the considerations of vaN DER Waars, Contin. II p. 41 et sqq. and p. 104, follows that the mixing of two substances in the fluid state is brought about in consequence of the molecular motion depending on the temperature 7’, and promoted by the mutual attraction of the molecules of the two components determined by the quantity a,,, whereas the attractions of the molecules of each component inter se determined by a,, and a,,, oppose the mixing. 1) Van Laar, These Proc. May '05, p. 38, cf. p. 39 footnote 1, treated the projection of the plaitpoint curve on the v, «-plane for such a mixture, without, however, further investigating the shape of the spinodal curve and of the plait. 2) The possibility of the occurrence of a longitudinal plait at temperatures above the critical ones of both components was supposed by van per WaAats in his treatment of the influence of the longitudinal plait on critical phenomena. (Zittings- versl. Kon. Akad. v. Wetensch. Amst. Nov. 1894, p. 133. [Added in the English translation |. ( 787 ) If the mutual attraction of the molecules of the two components ds is small compared with the attraction of the molecules of one of the components inter se, @,,, the appearance of complete miscibility will be determined solely by the molecular motion, and then the tempe- rature will have to be raised to an amount which, if some propor- tions of the /’s can occur then, may greatly exceed the critical temperature of the least volatile component, 7%, '), and with it the critical temperatures of all mixtures of these components. Thus from the equation (@) of vaN per Waats, Contin. IL p. 48, follows Tin =1.6875 7), for the eritical temperature of complete miscibility (VAN DER Waats l.c.) Vhs, if a = t= 0--and Db = h, 22 , may be put. At a lower temperature the two substances considered are only partially miscible, whereas for such a temperature above 7, there may be coexistence of two phases which, as will be further explained in $ 3 and 4, are to be considered as gas phases. Now it seems to follow from the nature of most of the substances known to us, most likely from the structure of their atoms, that Db, is also small, when a,, becomes very small; hence for a gas without cohesion 6,, may not be put equal to 4,, of a gas with cohesion, and as according to the equation cited of VAN DER WAALS a small value of 6,, furthers the mixing greatly, the critical tem- perature of complete miscibility cannot rise as high as was derived iust now. But though most likely the case mentioned just now as example does not occur in nature, yet it is certainly conducive to a better insight of what is to be expected for gases of exceedingly slight cohesion. § 2. The shape of the spinodal curves and the form of the plait on the wsurface for binary mixtures of which one component is a gas with molecules with extension and without cohesion. In fig. 1 Pl. I the spinodal curves are represented for such a case. The figure refers to the y-surface for the unity of weight of the mixtures, as we hope to give a further discussion of such a w-surface (comp. $ 1), also with a view to the treatment of the barotropic phenomena which may occur for these mixtures’) in case of a suitable proportion of the molecular volumes of the components, for which treatment the use of the y-surface for the unity of weight readily suggests itself. As was also mentioned in Comm. N°. 965, the conditions for 1) van DER Waars, in the paper cited p. 786 footnote [1], brought this in connec- tion with the great amount of heat absorbed at the mixing of such substances. [Added in the English translation]. 2) Cf. Comm. N°. 96a (Nov. '06), 96 b (Dec. ’06) and Mc (Dec. '06, Febr. ’07). waar +! ik ¢ ( 788 ) es coexistence may be studied by the aid of the y-surface for the unity of weight in the same way as by the aid of that for the molecular quantity ; moreover it is easy to pass from the former to the latter, which offers advantages for the treatment of many problems (cf. $ 6) if this is desired. The equation of the spinodal curve on the y-surface for the unity of weight of mixtures, for which Var per Waars” equation of state for binary mixtures with @ and 4 not depending on v and 7’ for constant « may be applied, and for which @,,=Wa,,¢,,, bou" O arbo) (ef. Comm. No. We, Dee. “06, p. 510) may be put, *) runs: R, R, Te = 2 R, (l—-e)ie a, —b‚, War + 2R, efo an — bv af. Here R, and R, are the gas constants for the unity of weight of the components concerned. For a,, = 0 this equation passes into: R 2 dro’ = (1—2z) [} 3 o — (1 — ep? + , ed, — «)] 3 12 * r e = . . . if we put Sih En “ef The roots of thìs equation in w have k Eh been determined by a graphical way for detinite values of 2 and tr. The figure has been construed for mixtures for which A/R, = */,, bb, = '/, (ef. Comm. N°. Me, Febr. “07, p. 600, footnote 2). With reference to Fig. 1 we point out that for 7< Tia (= 1.299 Ti) and >> 7: a spinodal curve closed on the side of the increasing v's, and together with it a similar plait, extends on the y-surface from the side of the small v's. At 7’= 7%, this plait reaches the side or the least volatile component. At lower 7’ the spinodal curve has two distinct branches, and the plait runs in a slanting direction from =~ the line re = 6 to the side of the least volatile component. Thus the investigation of mixtures with a gas without cohesion calls attention to a plait that starts from the side of the small volumes, and at lower temperature runs in an oblique direction to the side of the figure, which plait can be distinguished from the transverse and from the longitudinal plait. The spinodal curve for r= 1.040 has a barotropic plaitpoint Ps. (see Fig. 1), For 1.299 <“1r< 1.040 the angle with the v-axis of Tt the tangent to the plait in the plaitpoint *) 6,; >>. for 1.040<1r<1 is B. The barotropic phenomena for such a plait will be further 1) The quantities a, Gog, Ara, by, be, Dig, ete. relate to the unity of weight, @},¥, aam etc. to the molecular quantity. > Cf. Comm. N°. 965. ( 789 ) discussed in a following communication ‘ef. N°. 96e Febr. ’07, p. 660, footnote 1). In Fig. 2 the course of the plait has been schematically repre- sented for a temperature between the barotropic plaitpoint temperature and the critical temperature of the first component. The — - — -— curves denote the pressure curves, the — — — — curve the spinodal curve, the continuous curve the connode. The straight line AB is the tangent chord joining the coexisting phases A and B, CD is the barotropic tangent chord (Comm. 965). § 3. Limited miscibility of two gases. For mixtures where as in fig. 2 a plait giving rise to phases separated by a meniscus which coexist in pairs, represented in the figure e.g. by A and 5, while mixtures in intermediate concentrations are not stable, extends on the w-surface from the side of the small v’s at temperatures above the critical temperature of the least volatile component, we shall call not only the phase B a gas phase, for which it is a matter of course, but also the other A; so the latter may be called a second gas phase, and we may speak of equilibria between two gaseous mittures at those temperatures. That there is every reason to do so in the case treated in § 2 appears already from this, that the reduced temperature of the phase A, calculated with the critical temperature of the unsplit mixture with the concentration of A, is so high that already through its whole character the phase must immediately make the impression of a gas phase (so a second one). The shape of the p-lines in fig. 2 shows further, how the two coexisting gas phases may be obtained by isopiestic and isothermic mixing, in which nothing would indicate a transition to the liquid state, from the gas phases M and N of the simple substances *). We shall explain in the following § that it is really in accordance with the distinction between gas state and liquid state for binary mixtures in general, when we call A a second gas phase. § 4. Distinction between gas and liquid state for binary mixtures. It is true that since the continuity of the gas and the liquid state of aggregation has been ascertained, it may be said with a certain degree of justice that it is no longer possible to draw the line between the two states, but when in the definition of what is to be under- stood by liquid and what by gas we wish properly to express the difference and the continuity in the character of the hetero- geneous region and the homogeneous region and to preclude con- 1) Cf. footnote 1 p. 792. ( 790 ) clusions!) which are irreconcilable with the most obvious conception of phenomena, then the limits allowed for making this definition, are very narrow. Thus for a simple substance no other distinction will be possible than by means of the isotherm of the critical temperature, and the border curve (connodal curve on Gipps’ surface), which is divided into two branches by the eritieal state (plaitpoint of the connodal curve), of which the branch with the larger volumes is to be defined as gas branch, that with the smaller volumes as liquid branch’). Liquid phases are only those which by isothermic expansion may pass into such as lie on the liquid branch of the connodal curve, and also the metastable*) phases lying between the connodal and the spinodal curves, which may be brought on the liquid branch of the connodal curve by isothermie compression *). For binary mixtures the consideration of the w-surface of VAN DER Waats leads in many cases to definitions which are just as binding. 1) So Turesen’s definition, Z.S. für compr. und fl. Gase 1 (1897) p. 86, according to which e.g. strongly compressed hydrogen at ordinary temperatures would have to be called a liquid. 2) This is in harmony with the principle of continuity of phase along the border curve according to which a change of the character of the phases on a border curve can only occur in a critical point. For substances which at tempe- ratures near the critical one, in states represented by points on, or in the vicinity of that branch of the connodal curve on Grses’s surface which connects the liquid states at low temperatures with the plaitpoint, should be associated to mul- tiple molecules of which the volume is greater than the volumes of the composing molecules together, this principle would admit the possibility that on the liquid branch of the border curve liquid phases should occur with greater volume than the coexisting gasphase. Such simple substances would then show the barotropic phenomenon, tll now only found for binary mixtures. There is nothing known that points in the direction, of making the existence of such simple substances probable but there can be no more given a reason why it should be impossible. | Added in the translation]. 3) The metastable states have not been included in Bottzmann’s definition Gastheorie II, p. 45. 4) We do not accept the principle of the distinction of Lenman, Ann. d. Phys. 22 (1907) p. 474: “Erst die unterhalb der betrachteten Isotherme liegenden Kurven, welche in ihrem S-förmigen Teil unter die Abszissenachse hinunterreichen, ent- sprechen wahrer (tropfbarer) Flüssigkeit, d. h. einem Zustand, der negativen Druck zu ertragen im stande is”, as depending on the meaning that the existence of capillary surface tension in liquids which can form drops, would presuppose that these liquids can bear external tensile forces, i.e. negative pressures without split- ting up (cf. ibid p. 472 in the middle, and p. 475 at the top). [Added in the translation |. (791 ) When discussing this we shall leave out of account the case of solid states of aggregation and three phase equilibria. In the first place gas states are all the states on the y--surfaces on which there are no plaits. As criterion to divide states which belong to the stable or metastable *) region of w-surfaces which show plaits, into gas states and liquid states, analogy with the simple substance indicates their relation with the connodal curves of those plaits while for the metastable states the help of spinodal curves is to be called in. For this first of all the distinction between the two branches of the connodal curve of a plait is required. For in the first place we shall have to give the same name to each of the two branches of a connodal curve separated by one or two plaitpoints throughout its length ’). Now, on account of the existence of the barotropic phenomenon we cannot simply call gas branch of the connodal curve that at which one of the isopiestically connected states has the smallest density 3). It is therefore the question to indicate if possible on each branch a state whose nature is already known through the definition holding for simple substances or for those which behave as such when splitting up into two phases. In this different cases are to be distinguished. For the case that the considered plait‘) extends from one of the side planes «=O or «=1 over the w-surface, follows from the definition of gas phase and liquid phase of a single substance that the branch of the connodal curve from the gas state of the pure substance to the plaitpoint is to be called gas branch, and also that the branch from the liquid phase of the simple substance to the plaitpoint is to be ealled liquid branch. The gas branch and the liquid branch of the spinodal curve may be distinguished in the same way as those of the connodal curve. Let us restrict ourselves for the present to the distinction of gas and liquid in this case. In the first place we make use for this purpose of the isomignic (Comm. N°. 965) compression and expansion. 1) It follows from the nature of the case that unstable states have not to be considered here. 2) Cf. p. 790 footnote [2]. 3) Even it if we wish to leave gravity out of account, and pay only attention to the molecular volume of the phase, the barotropic phenomena have yet called attention to the possibility that we may find the gas volume first larger and then smaller than the liquid volume when passing along the same connodal curve. 4) The case of the two plaits at minimum critical temperature is comprised in this. ( 792 ) Every phase which cannot be brought on the connodal curve through this operation, or if it can, comes on the gas branch, will have to be called a gas phase, every phase which is made to lie on the liquid branch through isomignic expansion is a liquid phase. Besides the phases lying between the connodal and the spinodal curve which isomignically may be brought on the liquid branch of the connodal are metastable liquid phases. Besides the isothermic and isomignie compression without splitting there is another operation already mentioned in $ 3, which may help us to form an opinion about the similarity of different phases, viz. the isopiestie and isothermic mixing.') With regard to this phases which have been obtained by isopiestic admixing without splitting from phases of which it has been ascertained that they are to be called liquid phases, must be called liquid phases until in another way, (e.g. because no splitting takes place with isomignic compression and expansion) they have been proved to have passed into gas phases. *). Proceeding to the case that the plait from higher temperature appears as a closed plait on the w-surface, as long as the plaitpoint which first comes into contact with the side with decrease of tem- perature, has not yet come into contact, and with decrease of tem- perature the plait has not yet reached a mixture which on splitting behaves as a simple substance, and for which the distinction in liquid state and gas state is therefore fixed, we shall have to con- sider that branch of the connodal curve on the side of this plait- point, which passes into that of the gas phase at lower temperature, as belonging to the ordinary gas phase, whereas the branch which passes into the liquid branch at lower temperature may be looked upon as a second gas phase, and we are the more justified in doing so as the temperature should lie further above the critical tempera- 1) With the continuous isothermic and isopiestic mixing of two similar phases a and b the case may present itself (divided plait in the case of minimum crit. temp.), that an intermediate phase c of the other kind is obtained. So in general we cannot conclude to the similarity of c from the isothermic and isopiestic mixing of similar a and b. 2) This criterion is particularly of application to the retrograde condensation Qnd kind. For then phases on the connodal curve between the plaitpoint and the critical point of contact are liquid phases, phases on the p-curve through the plait- point and phases with the same x as the critical point of contact just the transi- tions to gas phases. The phases within the triangle bounded by these two lines and the connodal curve are also to be considered as liquid phases. Here we abstract from the small uncertainties which would be caused in these definitions when capillarity ought to be taken into account. | Added in the translation |. ( 798 ) tures of the unsplit mixtures belonging to the phases lying on them. Whereas in the case, that at a temperature comparatively little lower also the other side of the w-surface is reached by the origin- ally closed plait, the difference of the second gas phase with a liquid phase is still not very conspicuous, this may become very clear for the case of § 2, to which we have now got at last, that viz. with decreasing temperature a plait comes from the side » — 4, on the w-surface, and the plait appears for the first time as longitudinal plait. Now we may again call PSDF the branch of the first gas phase, PACE the branch of the second gas phase. It will certainly be obvious to speak of gas phases when a// the parts of the plait are found above the critical temperatures of the unsplit mixtures, and we shall decidedly have to speak of two gas phases, when the second branch of the connodal curve is intersected all over its length by isomignic lines on which beyond this plait no splitting up occurs, or if it is at most touched by one of them in the point » = 5. For then it is beyond doubt that the final point of that branch must be called a gas phase. Possibly also phases between the isomignic line of the critical point of contact, the line v= 4, and the second gas branch belong to the second gas phase. § 5. The surface of saturation for equilibria on the gas-gasplait. In fig. 3, 4 and 5 the sections 7’=— const. of the p, 7, x-surface of saturation for equilibria on the gas-gasplait have been schematically drawn for a mixture in which one component is a gas without, or almost without cohesion, in fig. 3 and 4 for temperatures higher than the critical temperature of the first component, in fig. 5 for this last temperature. In these figures too the division of a gas phase into two gas phases, and the transition of a part of the gas region into the liquid region at 7’= 7%, is clearly set forth. The — — — — curve is the locus of the plaitpoints. In a following communication, in which the properties of the w-surface for such mixtures will be further discussed, 7’, «sections ete. will be drawn of this surface of saturation. At the same time it will then have to appear in how far retrograde unmixing of a phase into two other phases is to be expected. That one of these phases may be called a second gas phase, appears in § 4. § 6. On the conditions which must be fuljilled that limited mis- ( 794 ) cibility of two gases may be expected. Now that it has appeared that on the suppositions mentioned in $ 2 for mixtures in which one component is a gas without cohesion with molecules with extension, limited miscibility might be expected in the gas state, the question rises whether this phenomenon is also to be expected for mixtures with a gas of feeble cohesion. As on the said suppositions no maxi- mum critical temp. is to be expected, this will be the case when Tim > Tr, is found.*) We have treated this question by the aid of the w-surface for the molecular quantity (cf. $ 2). We arrive then at the equations developed by van per Waats Contin. I p. 48. The condition that 7%, > 7), is: fi 2 4 Peony l {boon / bum — Vaz / aim}? > an 6? / Oat - 27 wy (l—ay) in which by and zy(l—ry) follow from the equations given by VAN DER WAALS loc. cit. We find from this’) Tin > Ti, for boom/bum=2 , Wt asoy/anm< 0.58 és 1 0.053 /, 0.0037 1, 0.00023 1, 0.000015 It appears on investigation that only for few pairs of substances the ratios of the as and /’s*) will be able to satisfy this condition. The still unknown relations between a and / for a same substance, to which we alluded in $ 1, and from which ensues that in general substances with small a also possess a small 4, and that as a rule large 5 goes together with large a, seem to prevent this. H e, which with a 6 which is still not very small compared with H, possesses a very small a, so feeble cohesion, and H,O, which taking the value of a into consideration, has a comparatively small 6, so a molecule of small volume, constitute exceptions to this general rule which are favourable for the phenomenon treated here. If for He — H,: boy / bum = ‘/,, and Aaa / Aum = NE (Comm. N°. 96e, Febr. ’07. p. 660 footnote 2), Trim < 7;, must be expected on the above suppositions. Also for helium-argon and helium-oxygen e.g. the same thing must be expected. Most likely the ratios are 1) Whether limited miscibility in the gas state may also occur if Tim < Ts, in certain cases and at suitable temperatures, will be discussed in § 7. 2) For baan) ban !/g eg. we find also Tem > Tr, for 0.125 > azaar / iim > 0.061. These cases will be further discussed. : 5) See e.g KOHNSTAMM, LANDOLT-BÖRNSTEIN-MEYERHOFFER's Physik. Chem. Tabellen. ( 795 ) more. favourable for mixtures of helium and neon *) than for those o. helium and hydrogen. For mixtures of helium and water the ratios for the above assumed aye and by. are such that for them limited miscibility in the gas state is to be expected, if the suppositions mentioned in § 2 are to be applied. The coefficients of viscosity and of conduction of heat (cf. Comm. N°. 96e, Febr. O07 p. 660 footnote 2) admit a value of bg. which is still somewhat though only little higher; this might render it possible to realise the said phenomenon perhaps also for the other pairs of substances mentioned, especially when we bear in mind that its appearance is not excluded for 7, < 77, (ef. p. 794 footnote igi’ The experimental investigation of these mixtures has been taken in hand in the Leiden Laboratory. (Communicated in the meeting of March 30). § 7. The shape of the spinodal curves and of the plaits for the case that the molecules of one component exert some, though still feeble attraction. With very small value of the mutual attraction a,, of the molecules of the two components, in connection with the feeble attraction a,, of the molecules of one component inter se, the spinodal curve will with decreasing temperature extend more and more on the y-surface as in Pl. I fig. 1 from the side of the small v's, come into contact with the line e=—O at T= 7},, and then cross from the line v ==b to the side c—0O in two isolated branches *). We leave here out of account what takes place at lower temperatures when the spinodal curve approaches and reaches the side z—1 too. To examine what shape the spinodal curve can have with greater attraction of the most volatile component, we shall avail ourselves of the suppositions introduced in $ 2 and also applied in $ 6 con- 1) Cf. Ramsay and Travers, Phil. Trans. A. 197 (1901) p. 47 for data con- cerning refractive power and critical temperature of neon. dT xpl 2) Here >0 fori z=0. We see here that VerscHArrett’s conclusion (These Proc. March 1906 p. 751) concerning the maximum temperature in the plaitpoint curve for mixtures, for which the component is indicated by a point from the region OHK (see fig. 2) must be supplemented by the possibility that he branch of the plaitpoint curve starting from the first component, goes to infinite pressures. ; 5d Proceedings Royal Acad. Amsterdam. Vol. IX. ( 796 ) cerning the equation of state and the quantities a,,*) and 6,,. In the net of spinodal curves for a given pair of substances 2 singular points may then occur, belonging to the spinodal curves for different temperatures. The values of « for these are determined by the equation : am 3m V aga = 2buum Paumaaom — beam V aim = — — (1) lem = buy Vase =F 2baem WP aypmavom + 3b6292MV aM For very small a,, we find from this two singular points with x >1, so not belonging to that part of the w-surface which can denote phases of mixtures. Of these two singular points that for which the lowest signs hold, passes through infinity for increasing Ass, and then approaches the line «=O on the other side of the y-surface. This line is reached for: PV asam/aum = oe 1+V14 3 bom/biim}—=m, . (2) With increasing a,,/a,, the singular point, which appears to be a double point for this region, approaches the line » —= 6, which line is reached for: Pam /atim = — (L—b2em biim) + W1—b20y1/bii m+ (been /bi an) == m, (3) In this we assume baam < bum). So if the mutual attraction of the molecules of the most volatile component and those of the other in connection with the attractions inter se attains a certain value — on the assumptions made for the calculations for m=V* asm/dnm=m, — the spinodal curve for Ll’ = T;, will no longer touch the side in K, (ef. fig. 1 PI. D, but it ) In this first investigation of what may be expected for mixtures of helium, with a view of forming some opinion as to the conditions under which the ex- periments for this purpose are to be made, we put (§ 2), bim =} (bum + baan) Um = V dimd22m (ef. Comm. Suppl. No. 8, These Proc. Sept. 'O4 p. 227) in the calculations, no data concerning a,, and b,, for those mixtures beinz available as yet. Also Van per Waats (These Proc. Febr. ’07, p. 630) assumes that as a rule diam < (dunt Azen). It will be necessary for a complete survey concerning the different possibilities to make also other suppositions about ayy (cf. VAN DER Waars le, Kounsramm ibid p. 642), at the same time taking care that a and b are not put independent of v and 7. at least not both (cf. Van per Waats, These Proc. Sept. ‘05 p. 289) and that they may only be put quadratic functions of x by approximation. If also for mixtures with very small a2, diem might be < V aumaom (ef. Konystaum le.) the phenomena of limited miscibility under discussion might still be sooner expected. *) For bx: > bum the other singular point comes from side z=1 on the J-surface for a smaller value of a22u/aimm. As probably this case does not present itself for the pairs of substances with small @9/a,, known to us, we shall not discuss it. ( 797 ) will have a double point there, in which the two branches of the spinodal curve intersect each other and the line «=O at an angle. In this case the critical temperature of the least volatile component is not changed in first approximation by small quantities of admixtures. With greater attraction of the most volatile component — on the suppositions mentioned for m,< m<c m, — a spinodal curve on the y-surface will have a double point. This will lie the nearer to the side of the small w’s, the more the attraction of the most volatile component increases. With a certain value of the attraction — m=m, — the spinodal curve reaches the line vy = 5 with a double point, with greater attraction the spinodal curve will proceed from a= O on the w-surface with decreasing 7’, and touch the line v=é at T= Ti. On the suppositions mentioned for b22mM/bim < ‘°/,, the contact with the line v=6 will here take place at temperatures > 7; , for baam/bum >7°/,, at 1 << Ti, so that in the latter case the spinodal curve comes first into contact with the line «= 1. In the first case (b22m/dim<c''/,,) a plait will come from «= 0 and at lower 7, whereas for larger m a branch plait directed to the side e—1 may develop: if m<cm, it will be united through an homogeneous double plaitpoint (KorreweG, Archiv. Neerl. 24 (1891)), with a plait coming from v= 6 to a plait that crosses from one side to the other, if m >> m, it wili pass into such a plait by contact with v= 6. In the second case the plait which becomes from «#=0O will again united with one coming from v = 6 for smaller m; for larger m a branch plait will have developed before this union takes place or before the spinodal curve touches the line v = 6. The shape of the spinodal curve for these cases with always greater attraction of the most volatile component, where we shall have to consider three phase equilibria, need not be discussed for the present, as they do not belong to the case of a component with feeble attraction *). For some values of baam/bijm table I gives the values asov/diim = mn, calculated from the equations (2) and (3). If we compare with this the values of d22m/diim for which Zn —= Ty, ($ 6) we see that they really lie between those calculated here. The shape of the spinodal curves for a case, in which m,<<Cm<m,, has been represented on plate II, for the w-surface of the unity of weight (cf. § 2), with the relations and data assumed in § 2, except that «,, a,, = 0.00049 (or doom/aiim = 0.00196). 1) Cf. moreover Van Laar, Arch. Teyter (2) 10 (1906), These Proc. Sept. ‘06 p. 226. ( 798 ) TABLE I. baam/Oum | m,* ae Vp 0.0014 0.0179 My 0.000134 0.000527 - Is | 0.000011 0.000022 The plait extending on the w-surface from v = 6 for a temperature > Ti, will have to be considered as a gas-gasplait according to §4 (ef. $6). Also a similar plait for 7’< 7%, if the connodal curve is not touched by an isomignic line, and is nowhere cut by an isomignie line which intersects the connodal curve of the plait coming from rz == 0". According to § 4 we shall be justified in considering also the plait lying on the side of the small v’s for 7, >7>Ta,i (temperature for which the double plaitpoint considered occurs) as gas-gasplait, if the temperature is above the critical temperatures of the unsplit mixtures for all parts of that plait. That there can be some reason for doing so, appears when we calculate the reduced temperature for the double plaitpoint for some cases, e.g. for the ratios basm/bijm and the m, belonging to it, mentioned in Table I. Putting bosm/bium =n the double plaitpoint temperature is deter- mined by : (nm) 27 TalTe, = — m(l oi Tha 4 cd (n+ m)’ and 2 m (n-+m)? 3 (1-+-m)? © m(2—m) — n(1—2m) So for the case represented on Plate II we find: wap = 0.587, Tapt/T ey — 0.966, Tayi [Ter = 2.17. (To be continued). Vdpl Vk, = 1) Here it appears that a gas-gasplait can occur also if Pkem < Tk,, and for temperatures T< Ts with Tkn > Tk,, (cf. p. 794 note 1 and p. 794). (April 25, 1907). H. KAMERLINGH ONNES and W. H. KEESOM. Contributions to the knowledge of the y-surface of VAN DER WAALS. XV. The case that one component is a gas without cohesion with molecules that have extension. Limited miscibility of two gases. Plate I. Fig. 1 ff fe / ; a Jeyst | 5 Eg = / / ‘ > e ‘ ao i Pe A a el 25 a Fig. 3 Fig. 4 Pig. 8 Proceedings Royal Acad. Amsterdam. Vol. IX. ty : Ë ke ad hae air LL VN a a? os Sr 1S ae KD eo “XI [OA ‘Ulepsajsury "*peoy [efoy ssuipaa.01g Ga): Glo 4 IL 9F®Id ‘sosvS OM} Jo AZITIQIOSIM poyYIMIIT ‘uorsuogxo SAVY JeYJ S9[NI9JOU YIM UOISOYOO JNOYZIA Svs B ST JUeMOdMOD suo yey} OSvO AUL “AX ‘STVVM UAC NVA JO ogejans-r 04} JO oBpojmoug o4} OJ SUOINYIIJUO) 'WOSTAN 'H 'M PU? SANNO HONITHAWV A 'H KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM, PROCEEDINGS OF THE MEETING of Friday April 26, 1907. OG (Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige Afdeeling van Vrijdag 26 April 1907, Dl. XV). GONE BN Se J. Boeke: “Gastrulation and the covering of the yolk in the teleostean egg”. (Communicated by Prof. A. A. W. Husrecur), p. 800. (With 2 plates). F. M. Jarcer: “On the influence which irradiation exerts on the electrical conductivity of Antimonite from Japan”. (Communicated by Prof. P. ZEEMAN), p. 809. B. van Trrcur: “On the influence of the fins upon the form of the trunk-myotome”. (Com- municated by Prof. G. C. J. Vosmarr), p. 814. (With one plate). L. J. J. Muskens: “Anatomical research about cerebellar connections” (3rd Communication). (Communicated by Prof C. WiNkrer), p. 819. S. L. van Oss: “Equilibrium of systems of forces and rotations in Sp4”. (Communicated by Prof. P. H. Scnours), p. 820. J. D. van DER Waats: “Contribution to the theory of binary mixtures”, III, p. 826. C. Lev: “Velocities of the current in a open Panama-canal”, p. 849. (With 2 pl.). A. A. W. Husrecur: “On the formation of red blood-corpuscles in the placenta of the flying maki (Galeopithecus)”, p. 873. 55 Proceedings Royal Acad. Amsterdam. Vol. IX. ( 800 ) Zoology. — “On gastrulation and the covering of the yolk in the teleostean egg.” By Dr. J. Borke. (Communicated by Prof. A. A. W. Husrecat). (Communicated in the meeiing of January 26, 1907). 1. Generally the process of gastrulation in teleosts is described by the greater part of the embryologists as a folding in of the margin of the blastoderm and the forming, partly by this process of folding and partly by delamination, of a mass of cells that contains the elements both of the chorda and mesoderm and of the entoderm. Only Wacraw Berent, M. v. KowaLewskr (in his paper of 1885), F. B. Sumner and myself have described a more or less independent origin of mesoderm and chorda on one side and the entoderm on the other side. Sumyer called the mass of cells lying at the posterior end of the embryo, from which the entoderm originates, prostomal thickening; I kept the same name for them and regarded these cells as being derived from the periblast. The large pelagie eggs of Muraenoids, which I could collect in large quantities at Naples, offer an extraordinarily good object for the study of these processes, much better than the eggs of Salmonides, studied chiefly by French and German authors’). The formation of chorda and mesodermie plates out of the folded portion of the blas- toderm, and of the entoderm out of the “prostomal thickening”, the mass of cells that lie at the hind-end of the embryo and are connected with the superficial layer and with the periblast, is clearly to be seen from the beginning of the formation of the embryo until the closure of the yolk-blastopore (confirmed by Sumner in his paper of 1904) and after a renewed careful study of these eggs *) I can only confirm entirely and in full the conclusions arrived at in my former, paper *) and the observations described there at some length. But in accordance with the new and better definition of gastrula- 1) Neither Henneauy, nor Korscu or JABLONOWSKI, to take a few examples, did see anything of these differentiations. Sumner gives however of Salvelinus very clear figures and descriptions. (Arch. f. Entwickelungsmech. Bd 17. 1903). 2) During the last 2 or 3 years Muraenoid-eggs seemed to have disappeared entirely from the Gulf of Naples. Now (summer 1906) I found them again in sufficient quantities. When comparing the different eggs with each other, it seemed to me that they belong to a still larger number of different species than I concluded in my former paper (9), and that there must be distinguished at least 10 different species of Muraenoid eggs in the Gulf of Naples. Dr. Sanzo at Messina came to the same conclusion. 5) Petrus Camper, Vol. 2, page 185—210 1902, ( 801 ) tion in vertebrates, given by Husrecut and Kriset and confirmed by a number of other embryologists, this process in the teleostean egg too must be revised and more sharply defined. In my former paper I was led to divide the process of gas- trulation into two “phases”, one by which the gut-entoderm is formed and one by which chorda and mesoderm are differentiated. But now I think the line must be drawn still sharper and the second phase must be separated entirely from the process of gastrulation sensu strictiori. According to the definition given by Huprecnt gastrulation is a process by which a gut-entoderm is differentiated from an ectodermic layer, and thus the germ consists of two distinct layers. The process of formation of chorda and mesodermic plates, which follows directly on the process of gastrulation proper (notogenese Huprecur) is a secondary complication of the process, characteristic of the vertebrate embryo. The most primitive mode of formation of the entoderm, according to Husrrcur, is by delamination and not by invagination. But after all it is chiefly the outcome, the formation of the two germ-layers, that is of interest. As soon as these two layers are formed and may be distinctly separated from each other, the process of gastrulation is finished. This is for example in amphioxus already the case at that stage of development, in which the gastrula is cap-shaped, the two layers (ectoderm and entoderm) are lying close against each other, the segmentation-cavity has disappeared, but the blastopore still extends over the entire breadth of the original blastula-vesicle. All the following processes until the closure of the blastopore (“Rückenmund”’ of HuBrrcur) are notogenesis and lead to the formation of the back (churda) and of the mesodermic plates and to the closure of the gastrula-mouth. When we study again the teleostean gastrulation-process from this point of view, we come to the conclusion, that in this case the process of gastrulation is ended as soon as the prostomal thickening has been formed, viz. at the beginning of the covering of the yolk. At that moment the ‘‘Anlage” of the entoderm is clearly differentiated, and the ectodermal cells begin to invaginate to form the chorda and mesodermic plates ; the concentration of the cells towards the median line begins, the long and slender embryo is formed out of the broad and short embryonic shield. The blastula-cavity, in the cases in which it is developed, has disappeared as such; all the following processes, the longitudinal growth of the embryo, the covering of the yolk by the blastoderm ring, the closure of the yolk blastopore, belong to the notogenesis and we are no more entitled to reckon these processes 55 ( 802 ) to gastrulation proper as we are to do that of the covering of the yolk by the entoderm in sauropsids. During this longitudinal growth of the embryo new cells are produced by the prostomal thickening and pushed inwards to form the entoderm, but this may not be called gastrulation any more. The period of development, during which the yolk is being covered by the blastodermring, differs much in different embryos. In muraenoids at the time the yolk-blastopore is closed the embryo possesses from 5 to 10 pairs of primitive segments; the issuing larvae possess 58 to 75 segments. In salmonidae at the closure of the yolk-blastopore of the 57 to 60 segments 18 to 28 are differentiated. The other organs too are developed to a greater or lesser degree. To use the term gastrulation for the processes during this whole period of development leads us into difficulties. Tue first question we have to answer, when we study closer the process of gastrulation in teleosts, is: at what time does the process of gastrulation begin in the large meroblastic eggs? Recently Bracuer') has called attention to.a process, which he calls “clivage gastruléen”, and which he describes for the eggs of Rana fusca as the formation of a circular groove at the base of the segmentation-cavity around the yolk-mass, before there is to be seen a trace of a blastopore (Rusconic groove) at the outside of the egg: ‘immeédiatement?) avant que la gastrulation ne commence, la cavité de segmentation, sphérique ou a peu pres, occupe l’hémisphere supérieur de l’oeuf (de Rana fusca)... Bientôt, sur tout le pourtour du plancher de la cavité de segmentation, une fente se produit par clivage; cette fente ’s enfonce entre les cellules de la zOne marginale et les divise en deux couches: Pune, superficielle, prolonge directement la voûte de la cavité de segmentation, mais est formé par des cellules plus volumineuses et plus claires qu’ au pôle supérieur; Pautre, profonde, fait corps avec les éléments du plancher. Cest ce clivage, que j'ai appelé “clivage gastruléen”, c'est lui, qui caractérise la premiere phase de la gastrulation, parce qu'il amene, en dessous de Véquateur de loeuf, la formation d'un feuillet enveloppant et d’une masse cellulaire enveloppée, d'un ectoblaste et d'un endoblaste.” And farther on: “lorsque ce clivage est achevé, il est clair, qu’a sa limite inférieure, lectoblaste et Vendoblaste se continuent lun dans l’autre, comme le faisaient antérieurement la votte et le plancher de la cavité de segmentation.” This line of continuity Bracner calls “blastopore virtuel” ; after a short time this virtual blastopore is converted into a real blastopore 1) Archives de Biologie Tome 19 1902 and Anatom. Anzeiger. Bd. 27 1905. 2) Anat. Anzeiger Bd. 27, p. 215. ( 803 ) by the formation of the groove that leads to the formation of the archenterie cavity. This groove is formed by delamination; until now there is no trace of invagination. This begins in what Bracuer calls the second phase of the gastrulation process, which leads to the formation of the archenteric cavity in its entire width, and is synchronic with the process of notogenesis, of the formation of the back of the embryo; “quand les lévres blastoporales se soulèvent, quand de virtuelles elles deviennent réelles, c'est que le blastopore va commencer a se fermer, c'est que le dos de lembryon va commencer a se former” (le. 1902, p. 225). Bracner is right here. Also there, where be draws a sharp line between the entirely embryogenie blastoporus of the holoblastic eggs and the blastoporus of the meroblastic eggs with a large amount of yolk, which is divided into two parts, an embryogenic blastoporus and a yolk-blastoporus. But when he reekons these processes, which occur in the selachian and teleostean egg during the covering of the large mass of yolk and the closure of the blastopore, still to gastrulation, when he calls the entire process of covering of the yolk “clivage gastruléen”, and calls the whole blastoderin ring “blastopore virtuel”’, he goes too far, and forgets the significance of the phenomena, oecurring at the end of segmentation and during the formation of the periblast. For the answer to the question, at what time does the gastru- lation in the teleostean egg begin, his analysis of the phenomena of this process in the amphibian egg is extremely interesting. The segmentation of the teleostean eggs is not regular during all its phases. When we combine the very accurate observations of Korscn on this account, we see, that in the segmenting blastoderm at a definite moment, about that of the 10" division of the embryonic cells, there occurs an important alteration. Until the end of the 10% cell-division (in Belone) the different cells divide wholly synchronic; in Torpedo Rickert found synchro- nism until the 9' division. By the tenth division the yolk-sac ento- blast is formed (in Gobius, Crenilabrus, Belone), the two nuclei of the marginal segments, resulting from this division, remaining in the undivided protoplasm; where this does not occur at the tenth division the deviation is very small (in Cristiceps argentatus it partly begins at the 9 division, in Trutta fario at the eleventh division). Syn- chronically with the differentiation of yolk-sac entoblast the super- ficial layer (‘“Deckschicht’’) is differentiated. At the end of the 10: division all at once the blastoderm alters its form: it gets higher, more hill-shaped and the diameter is lessened; the mass of cells ( 804 ) concentrates, the superficial layer is still more clearly visible as a definite enveloping layer of cells. It is just the synchronic differen- tiation of the superficial layer, which shuts off the blastoderm from the surrounding medium and is the only way by which the developing cells may get the oxygenium from the perivitelline fluid, on one side, and of the periblast, by means of which the blastoderm is nourished by the yolk, on the other side, which seems to me to be important ; by this synchronic differentiation a new phase in the developmental process is initiated, and the series of changes have begun that lead to gastrulation. Very soon the blastoderm-dise flattens, at first only because the superficial layer contracts a little, and the blastoderm sinks a little into the yolk-sphere (fig. 8) but after that because the blastodise itself spreads out, flattens (fig. 9). The cells come closer together, and soon the unilateral thickening that forms the first outwardly recognisable beginning of the building of the embryo, becomes visible. During these changes it is of no account whether a blastula-cavity is formed, or not. As I have described elsewhere, in different murae- noids during this stage a distinct blastula-cavity is formed, which may be seen in the living egg. Afterwards follows the flattening of the blastodise and the disappearance of the cavity as such. The closer study of young stages of the eggs of muraena N°. 7 *) showed me however, that in these eggs no blastula-cavity is formed, and that in this case the blastoderm, that takes just the same conical shape as the hollow blastoderm in the other muraenoid eggs, remains solid and is built up of a mass of loosely arranged cells. The further development is the same as in the other series (c.f. fig. 1—3 on . plate 1). This flattening of the blastodise, following on the stage just described, coinciding with the concentration of the cells of the blastoderm towards the side where in later stages the embryo is formed, and coming before the invagination (and partial delamination) of the blastoderm cells, that leads to the formation of the chorda and the mesodermic plates, is already a part of the gastrulation process and must be compared with the “clivage gastruléen” of the amphibian egg. Immediately on this ‘“‘chvage gastruléen” follows the formation of the prostomal thickening (that is the “blastopore réel” of BRACHET), there where the superficial layer or pavement layer is connected with the periblast, out of the surperficial cells of the periblast *) (c-f. 1) Comp. Perrus Camper, Vol. Il p. 150. 2) Sumner (l. c. page 145) saw evidences for this mode of origin in the egg of Salvelinus, but not in that of Noturus or Schilbeodes. On these two forms I can- ( 805 ) fig. 4. 5 and 6 on plate 1). It seems probable, that at least in somè cases entodermeells are formed by delamination from the periblast at some distance from the surface in front of the prostomal thickening (fig. 5e). So here, as in many vertebrates, the entoderm is formed by delamination. At the moment of the differentiation of the pros- tomal thickening (figs. 2, 4), there is still no trace of the invagination of the mesodermeells, only a thickening of the mass of cells lying just overhead of the cells of the prostomal thickening. Immediately afterwards however a distinct differentiation of the mesoderm becomes visible. At that stage the notogenesis begins and the gastrulation process „is finished. The prostomal thickening is the ventral lip of the “blastopore réel” of the Amphibian egg. For the developmental processes following on this stage I can contain myself with referring to my former paper. That here only a small, not very prominent tail-knob is formed and no far-reaching projecting tail-folds appear, as in the selachean embryo, is caused by the relation of the pave- ment-layer to the blastoderm and the periblast, which inflaence the development of teleostean egg (“développement massif” of HeNNeGur). 2. To determine the direction of growth of the blastodermring during the covering of the yolk, I used in my former paper the oil-drops in the yolk of the muraenoid eggs as a point of orientation, on the contention that these oildrops maintain (in the muraenoid egg) a constant position in the yolk. On this basis I constructed a scheme of the mode of growth of the blastoderm in the yolk. *) Both Sumner and Kopscu rejected this contention and the scheme, SUMNER because of the fact, that by inverting the egg of Fundulus heteroclitus in a compress, the oil-drops may be caused to rise through the yolk and assume a position antipodal to their original one, which shows, that here the oil-drop may not be regarded as a constant point of orientation in the egg. In this SUMNER is perfectly right. Not only in Fundulus, but in several marine pelagic eges too the oil-drops may be seen travelling through the yolk by converting the egg or bringing the young larva (in some species) in an abnormal position. In the muraenoid egg however the case is entirely different. Here the structure of the periblast and of the not judge, but I will only mention here, that the figures, drawn by the author, are taken of much too late stages of development, to be convincing. And after all, where the blastodermcells are so much alike, as is the case in most teleostean eggs, one positive result in a favourable case as is offered in the muraenoid egg, is more convincing than several negative results in less favourable eggs. ) 1, ce. page 142, ( 806 ) yolk-mass, which I described at full length in my former paper, completely checks the displacement of the oil-drops. This is to be concluded already from the behaviour of the normal egg. So in the eggs of Muraena No. + a large number of rather large oil-drops are lying at about equal distances from each other at the surface of the yolk-mass. During the entire process of covering of the yolk, the distance of these oil-drops remains the same, they maintain their relative position absolutely, and only during the slight disfigurement of the yolk-sphere, caused by the contraction of the blastodermring during the circumgrowth of the yolk (fig. 4 on plate 2) the position of the oil-drops is changed a little, only to become the same as before, after the yolk has regained its spherical form. When these oil-drops were lying loose in the yolk or in the periblast, they would have crowded together at the upper pole of the egg, or at least their relative position would have undergone a change during the covering of the yolk. Only when the yolk-mass in the developing embryo becomes pear-shaped and very much elongated (l.c. plate 2, fig. 6, 7), the oildrops of course change their position. Even then, however, they remain scattered through the yolk. Experiments also show the constant fixed position of the oil-drops in the muraenoid eggs. When we transfix the egg-capsule carefully with a fine needle, it is possible to lift one of the oil-drops or a small portion of the peripheral yolk out of the egg. The other oil- drops retain their normal position, and in most cases such eggs develop normally and give rise to normal embryos. When we operate very carefully under a low-power dissecting-microscope, it is possible to leave the oil-drop connected with the periblast by means of a thin protoplasmatic thread. When we do this in a very early stage of development, at the beginning of the gastrulation-process, we see that this oil-drop, which surely may be regarded as a fixed point on the surface of the egg, retains its position in relation to the other oil-drops, until it is cut off from the periblast by the growing blas- todermring. In fig. 2a, 2b, 2c and 2d on plate 2 I have drawn from life several stages of this process in an egg of Muraena No. 1. During my stay at the Stazione Zoologica at Naples, in August and September 1906, I performed several of these experiments with different muraenoid eggs. They all led to the same result, and con- firmed my former statements. And so I believe that my contention was right and that the scheme I figured is a true representation of the facts. Of course only in a general sense, for there are many individual variations (so for example the case figured in fig. 3 on plate 2). And after all, when we compare this scheme with that ( 807 ) given by Korsen for the trout, we see that they do not differ so very much, and that the displacement of the hinder end of the embryo is almost the same. In the text of my former paper however I expressed myself rather ambiguously, and brought my view into a too close contact with that expressed by Orrvacuer. The figures however show that my scheme differs rather much from that ot OBLLACHER. ‘But I differ from Kopscn in his supposition that the hea d-end of the developing embryo is a fixed point on the periphery of the egg. I find myself here quite in harmony with SumNur, who draws from the large series of his extremely careful and exact experiments the conclusion, that “the head end also grows, or at least moves, forward, though to a much smaller extent” (l. ¢. page 115), and says: “I regard it as highly probable (see Exp. 1, 3, 34, 35, 36 and Fig. 32) that the primary head end grows — or is pushed — forward from an original position on the margin” (l. c. page 139). From different experiments of the author we may draw the conclusion, that in many cases this forward growth of the head-end is rather extensive (exp. No. 6, 10, 11 (partly), 26, 35, fig. 10), and experiment N°. 6 (table VIII) among others shows, that under circum- stances the direction of growth may be entirely reversed, so that the tail-knob of the embryo remains at the same place, and the head-end bends round the surface of the yolk. Kopscu too, in his paper: “Ueber die morphologische Bedeutung des Keimhautrandes und die Embryobildung bei der Forelle”, describes an experiment with simular results in the trout. So it is not unreasonable to suppose, that in the spherical egg of the Muraenoids during the covering of the yolk the head end of the embryo is moving forward, and to a certain extent follows the growing blastodermring, which is the case chiefly during the later stages of the covering of the yolkmass, as I showed in my scheme. During the first stages of development it is chiefly the tail-end of the embryo which travels backwards, (see the scheme in my former paper and fig. 1—-3 or plate 2), and Kopscu is right to locate here the centre of growth of the embryo. The conclusion of Sumner, that for some time prior to the closure of the blastopore, the ventral lip of the latter (former anterior margin of the blastoderm) travels much faster than the dorsal lip (l. e. p. 115) is quite in harmony with my results tor the murae- noid egg described in my former paper. *) 1) Petrus Camper, I. c. p. 196. ( 808 ) 3. At the end of the covering of the yolk, at the closure of the blastopore, Kuprrrr’s vesicle is formed after the manner described at length in my former paper. By Swarn and Bracner’) and by Sumner the narrow passage connecting this vesicle with the exterior, through the closing blastopore, is regarded as representing the neuren- teric canal. I do not think they are in the right here. Kuprrer’s vesicle is a ventral formation. Dorsally it is separated from the cells of the medulla by the cells of the prostomal thickening and the pavement layer. An open canalis neurentericus is not found even in these forms. Kuprrer himself called the vesicle allantois. Husrecnut followed him in this. In my former paper I compared the vesicle with the allan- tois of amniota on physiological grounds, and I think it is a very good thought of Huprecur to take up the old name of Kuprrer and compare the vesicle with the allantois on morphological grounds. DESCRIPTION OF FIGURES ON PLATE 1 AND 2. Plate 1. Figg. 1—4. Median sections through eggs of Muraena N°. 1 on different stages of gastrulation. In fig. 3 gastrulation is finished and notogenesis is begun. In fig. 2 the structure of the yolk is drawn. Enlargement 40 times. Fig. 4a, 5 and 6 give median sections through the developing prostomal thickening and adjoining parts, seen under a higher power. Figg. 7—9. The flattening of the blastodisc at the beginning of gastrulation in eggs of Muraena N°. 7. Enlargement 4) times. Plate 2. All the figures on this plate are drawn from life as accurately as possible. Fig. la—le. Covering of the yolk in an egg of Muraena N. 1. Fig. 2a—2d. Covering of the yolk and closure of the blastopore in an egg of Muraena N°. 1. By means of a fine needle one of the oil-drops is nearly severed from the surface of the yolk, remaining connected with the periblast only by means of a thin protoplasmatic thread. In fig. 2¢ this oil-drop is cut off from the surface of the egg by the travelling blastodermring and is lying close against the egg- capsule EK. In fig. 2d (closure of the blastopore) this oil-drop is no more drawn in the figure. Fig. 3. Unusually fargoing dislocation of the hinder end of an embryo during the covering of the yoik. The head end lies approximately at the former centre of the blastodisc. Fig. 4. Compression of the yolk-sphere by the growing blastodermring in an egg of Muraena N°. 4. The oil-drops only temporarily changed their relative dis- tances a little. OD = oildrop. pv = prostomal thickening £ per = periblast. Bl = blastoderm. D = pavement layer = entoderm e Leiden, 17 January 1907. 2) Archives de Biologie T. 20. 1904, page 601. J. BOEKE. On Gastru Fig. 3. Fig. 4. J. Boeke del. Proceedings Royal Acad, Amsterdam. Vol, IX, J. BOEKE. On Gastrulation and the covering of the yolk in the teleostean egg. PLATE 1 Fig. 8. Fig, 6. Fig. 7. Fig. 9. J. Boeke del. Proceedings Royal Acad. Amsterdam. Vol. IX, J. BOEKE. On Gastrulation and the covering of the yolk in the teleostean egg. PLATE 2, Fig. 20. Figs 26 Fisk 2d, Fig. 3. Fig. 4, J. Boeke del. Proceedings Royal Acad, Amsterdam. Vol. IX, ( 809 ) Physics. — “On the influence which irradiation everts on the electrical conductivity of Antimonite from Japan)”. By Dr. F. M. JArGER. (Communicated by Prof. P. ZppMaN). (Communicated in the meeting of February 23, 1907). $ 1. Having been occupied for a considerable time with the determination of the specific electrical resistance in the three erys- tallographie main directions of the antimonite from Shikoku (Japan), I had already found that with this substance, which belongs to the very bad conductors, inexplicable irregularities presented themselves, when the resistance was determined several times anew during a long time, with identical electromotive force. Generally the obtained deflection of the galvanometer first became larger and larger, and decreased again in course of time, after which, as I found, periodical increase and decrease sometimes followed. It was impossible to detect any connection between tension, intensity of current, and time. As for rods of a length of some centimeters and a section of about a quarter of a square centimeter, resistances were found in the different directions lying between 500 and 20000 millions of Ohms, I first thought of an impregnation of the electrical charge in the ill-con- ducting material. On account of its opposed direction, however, an eventual polarisation current would have to cause an apparent increase of the resistance, whereas experience generaliy showed a decrease of the initial resistance. § 2. While I was trying to ascertain the cause of these deviations, a sunbeam fell through an aperture of the curtain on the piece of mirror-glass which closed the THomson-galvanometer, and was partially reflected to the apparatus containing the piece of antimonite, cut with its longitudinal direction parallel to the crystallographic b-axis, The needle of the galvanometer deflected immediately towards that side in which the total deflection was increased. At first I thought that the heat of the sun penetrating the galvanometer on one side had changed the cocoon thread so much as to cause a torsion. Some moments later, however, when I happened to light a match in the neighbourhood of the preparation, the imerease of the already existing deflection was repeated, and now in the same sense as before, and at the same time stronger. § 3. So we have met here with a new phenomenon. Either the ( 810 ) radiation of light, or the heat must be the cause of the phenomenon. I then undertook the following set of experiments. A rod of antimonite quite covered by paraffin, and cut parallel to the $-axis, was shunted into the circuit of a dynamo, the tension being kept at exactly 35 Volts by means of a resistance of incan- descent lamps. When shunting in the THomson-galvanometer *), which had been hung up in an antivibration apparatus of JuLius, and which was so sensitive, that at a distance of mirror of two meters, it still gave a deflection (double) of 26,5 mm. for a current of 0,000000006 Amperes, — we obtained a constant, single deflection of 10,7 cm. on the left of the zero point. An incandescent lamp (of 110 Volts), placed at about 2 meters’ distance from the preparation, gave an increase of this deflection of 4+ m.m., i.e. 3,7 °/, — agreeing in this case with a decrease of resistance of about 53 millions of Ohms. When the same lamp was placed at 1 meter’s distance it brought about an inerease of the deflection of 11 m.m.; at */, meter’s distance of about 20 m.m., and held near the rod for a short moment, of more than 220 m.m., i.e. an inerease of the conductivity of resp. EON ae 18,7 °/ and: 206"/,!*) Then the lamp was removed, and after the deflection had resumed about its original value, one of the curtains at the window was drawn aside, so that the diffuse daylight (overcast sky) fell on the apparatus. Instantly the deflection was increased by more than 4 m.m. i.e. about 3,7°/,. Then a wooden box was placed over the apparatus, and then removed. Every time the experiment was repeated the constant deflections in the light were found from 3 to 8 m.m. larger than those in the dark. § 4. In the foregoing experiments only exceedingly little light fell on the rod of antimonite, as it was quite covered by a coat of paraffin *) about 0,4 ¢.m. thick, and so only the light penetrating the half transparent coating could have any effect. Then the experiment was repeated as follows. A lamella‘) of antimonite was clasped between two much larger copper plates, which two plates were well insulated. The condensator (fig. 1) obtained in this way was suspended on silk threads. *) 1) The internal resistance of this instrument amounted to 6681 Ohms. 2) The resistance of the rod was diminished by an amount of more than 950 millions of Ohms in the latter case. 3) The purpose of these precautions will be explained later on in a paper written in conjunction with Mr. Vas Nunes. 4) The antimonite splits perfectly // (010), so y b-axis. ( 811 ) The antimonite plate had a thickness of about 1 m.m. and a section which may have amounted to about '/, ¢.m’. Now if a source of light was placed at I, the remaining deflection of about 1,8¢.m. obtained at a 10,5 Volts’ tension was only increased by 2 m.m., i.e. by about 11°/,. If, however, the light was placed at the same distance in U, the increase amounted to about 11,5 m.m., ies GELS One ae eee ee al are EE E.H In the former case the plate A is viz. in the shadow of K,, and so receives but very little light reflected by the walls; in the latter case, however, the radiation is direct. §5. If a thick plate of colourless plate-glass is placed between the source of light in If and the apparatus, the remarkable fact presents itself that the deflection is considerably decreased. The explanation of this phenomenon was obvious. For a copper bar, heated to some hundreds of degrees, and brought near the apparatus, immediately diminished the obtained deflection greatly. Hence — and this is a most remarkable result — rise of temperature has an influence directly opposed to that of radiation of light: it enlarges the resistance instead of diminishing it, as rays of light do. ( 812 ) ed If the plate is again removed, the deflection decreases again to the value it had before the plate was placed between etc. So this fully proves that it is the radiation of light which influences the conductivity of antimonite in so high a degree, and „ot the heat; for the latter diminishes the conductivity, in contrast with the former. § 6. Finally glass plates of different colours were interposed between the source of light and the rod. It then appeared that the influence of the differently coloured light was very different. The antimonite namely proved to be subjected to hardly any change by green light; for red light the increase was pretty large, for yellow light a little more, for green very small, for violet light again stronger. In each of these cases the deflection appeared to have resumed its original value after removal of the source of light’). With violet radiation I obtained an increase ot conductivity which amounted to about 150 °/, of the original value; with white light with interposed glass plate one of about 250 °/,. To get some insight into the quantitative action for a special case, the following experiments were made. An ordinary electrical incan- descent lamp was adjusted at 16 ems’. distance from an antimonite rod covered with a coat of paraffin 1 em. thick. First of all it was ascertained that action of light by itself did not excite an electrical current. It then appeared that the deflection of the galvanometer was increased just as much irrespective of the direction of the current. So the decrease of resistance is independent of the direction of the current. By interposition of coloured glass plates I got a rough estimation of the relative influence of the different colours of the spectrum. Thus I found: White light, placed at 16cems’. distance, makes the conductivity rise to 2000, of its original value Red EE ” Pye EE ” ” EE 9 14% „ 4 ” ” Orange ” ” $91.7 0 ER ” EE ” 9 19 158% +5, 55 ” 7 Green ,, ” 59905 39 ss EN ” ” sos LEGG oy Gs ” ” Blue EE ” 9959 2 ” ” EL ” ss 176% …„ 45 ” ” 1) Not quite the original value. The substance shows hysteresis to a certain amount, which, however, is smaller than for selenium. Already 20 a 40 minutes after the source of light had been removed, the original deflection was sometimes found back. The mineral seems to be quite free from any admixture of selenium, as a qualitative investigation taught me. Remarkable ina high degree is the fact, that on melting the natural mineral, it obtains, when solidified as a conglomerate of little crystals, a specific resistance, which is many thousand times less than before, while at the same time it has lost its sensibility to light- radiation quite. On heating the antimonite however, without melting, it conserves this property. Analysis has taught me, that there are present the elements: Sb, S, Ca, Ba, Sr, Si and, as Prof. Krey found, traces of Zn and Co; also Si03-erystals are included. (Added in the English translation). (.. Shs. } As heat-rays have only an exceedingly slight effect, and, as I ascertained later on in conjunction with Mr. Vas Nunes, also the ultraviolet light emitted by cadmium poles causes only a small increase of the conductivity, the dependence of wave-length and decrease of resistance is evidently represented by a curve whose minima lie in the ultrared, in the green and in the ultraviolet, and whose maxima are situated in the red and in the blue part of the spectrum *). Later on when the determinations of the resistance of this sub- stance will have led to favourable results, we shall make some closer communication on the relation between thermal and electric motion in this conductor. § 7. The phenomenon discovered here reminds strongly of that observed for selenium *). It is, however, noteworthy, that though the dependence of the increase of the conductivity on the radiation of light, and even on the wave-length of the light manifests itself in a perfectly analogous way to that for antimonite, yet the two differ in some respects. First of all for the selenium polymorphous changes, and. the displacement of equilibrium attending them play an im- portant part; then, however, the resistance always decreases here with rise of temperature, so exactly the reverse of what happens in my investigations, in which moreover there is no question whatever of polymorphous changes, as far as is known. An analogy between the two cases is to be found in the fact already discovered by Apams*), that the resistance decreases with rising electromotive force, also after continued action of it; such a deviation from the Jaw of Onm is also found for the antimonite. On the other hand the behaviour of antimonite from Japan seems to present a closer analogy with that of the crystallised te/lurium ; 1) Though it is not intended as an explanation, I will yet call to mind that it follows from Mürrer’s investigation (N. Jahrb. f. Miner. u. s. w. Beil. Bd. 17, 187— 251) on the optical constants of the antimonite from Japan, that the indices of refraction ny and #3 have their maximum values exactly for the green rays (between the lines E and F) (viz. m =5,47—5,53 and n, = 4,52—4,49), while also the double refraction reaches its maximum value for these rays. The polarisation of the reflected rays is right-elliptic (negative). However, on using polarized light, we could not find any influence of the direction of vibration: the variation of the electric resistance was in the two cases the same. (Added in the English translation). 5) G. Wiepemann, Die Lehre v. d. Elektricität. (1882). I. p. 544—553). 3) Sate, Phil. Mag. [4]. 47. 216. (1874); Pogg. Ann. 150. 333; Chem. News. 33. 1. (1876). 4) Apams, Phil. Trans. 157.; Pogg, Ann. 159. 621. (1876), Phil. Mag. [5]. 4. 115 ( 814 ) here, too, the resistance increases with heating, decreases with exposure to light *). In conjunction with Mr. Vas Nunes I hope shortly to publish also some quantitative data on the phenomenon discovered by me, and also on the behaviour of the melted and again solidified antimonite and the analogous selenium compound. This investigation has been made in the Physical Laboratory at Amsterdam. Anatomy. — “On the influence of the fins upon the form of the trunk-myotome’. By B. van Tricut. (Communicated by Prof. G. C. J. Vosmarr). (From the Anatomical Institute at Leyden). (Communicated in the meeting of March 30, 1907). This research forms a direct sequel to Professor LANGELAAN’s work “On the Form of the Trunk-myotome”, and is intended to show the influence of the fins upon the form of the myotome. The method which I followed, was based upon the chief result of the foregoing research viz. that the differentiation of the myotome takes place in a continuous manner by means of folding, and that it is possible to follow the process of folding in dissecting the intermyotomal tissue. Now the method of direct dissection proved to be restricted in its application, so that it was necessary partly to apply a more indirect one. This latter method rests upon the relation, which exists between the form of the myotome and the structure of the transverse sections of the animal. Differentiation of the dorsal musculature. From a rather large sample of Mustelus vulgaris the skin with the underlying connective tissue was removed, so that the external surface of the myotoms was laid bare (figure I). Then in the region before the first dorsal fin the parts constituting one and the same myotome were determined; the form of this myotome exhibited about the same form as in Acanthias, only the lateral part of the myotome proved to be displaced caudally; the breadth of this displacement amounted to about half the breadth of the myo- tome. This myotome was arbitrarily indicated by the number 1) | have made an arrangement with Mr. J. W. Gitray at Delft with regard to the mounting of antimonite preparations, and the preparation of antimonite cells for practical use. ( 815 ) 1 and the following myotoms by subsequent numbers. After that, transverse sections of the animal were made, of 1—2 em. thickness, and the numbering transferred to these sections, so that the lamellae belonging to one and the same myotome received the same number. For the sake of an easy description, figure II] gives a hemischematic representation of the myotome, in which the peaks are indicated by numbers, the lamellae by letters. In figure III which is the trans- verse section, (indicated in figure I with an A) the peaks appear as systems of concentric lamellae marked in accordance with the marking in figure Il. If we now pass to the region of the first dorsal fin (figure IV, section B of figure I and fig. V, section C of fig. D the image of the transverse section is changed, instead of being com- posed by four peaks, the dorsal musculature only shows three peaks. The peak, indicated as number 1, has disappeared and instead of this peak we find the first lamellae of the dorsal fin. Now in all subsequent sections this first peak does not reappear any more. By the method of successsive numbering it was possible to determine the first myotome losing its most dorsal peak (number 1). The external surface of this myotome is blackened in figure I. From the principle laid down in the beginning of this notice ensues, that the myotome apparently losing its first peak, gives a muscular element to the dorsal fin; this element is therefore also blackened in figure I. It may be seen in figure I that the first myotome giving an element to the fin lies a little caudally in respect to the front edge of this fin. The number of myotomes giving a part to the first dorsal fin may easily be determined, because these composing parts of the fin are separated by intersegmental tissue; that we have really to deal with intersegmental tissue follows from the fact that through these lamellae bloodvessels and extremely fine nerve fibres reach the skin (vid. v. Bisselick “On the Innervation of the Trunk-myotome’’). The total number of muscular elements composing the fin, amounted to 34, so that the last myotome still giving an element to the first dorsal fin already lies in the region of the second dorsal fin. From the fact that the most dorsal peak (number 1) does not reappear any more in the transverse sections, it follows that the next myotome gives the first element to the second dorsal fin. The surface of this myotome is also blackened in figure I to show its position in relation to the front edge of the second dorsal fin. It is evident, that this myotome occupies the same position in respect to the second dorsal fin as the first myotome does in respect to the first dorsal fin. The number of myotomes composing the second dorsal fin amounts to 30. Upon the second dorsal fin follows the dorsal part of the caudal 56 Proceedings Royal Acad. Amsterdam. Vol. LX. ( 816 ) fin. In this fin the myotomes are pressed together so closely that a direct counting of the number composing the fin is no more possible ; by comparing the total number of vertebrae to the number of myotomes composing the first and second dorsal fins, we find that about 70 myotomes give an element to the dorsal part of the caudal fin. The results obtained by this indirect method are corroborated by the result of the direct dissection. If we take a myotome giving a muscular element upon the more anterior part of the dorsal fin and begin the dissection with lamella 6 in the neighbourhood of the second peak and proceed preparing caudally, we find lamella 6 being rolled in, towards the mesial plane of the body, in the shallow excavation in which the base of the fin rests, (tig. VI‘. Along this way the muscular tissue becomes gradually atrophic and only a thin band remains, consisting of the connective tissue which forms the fraane- work for the muscle fibres. In the neighbourhood of the sagittal plane of the body this lamella is folded, in such a manner, that the line of folding (figure VI L" L") runs parallel to the sagittal axis of the body. By this process of infolding the direction of the lamella 6 is reversed, the infolded part proceeding cranially; this part of lamella 5 passes into the dense sheath of connective tissue, which is interposed between the dorsal musculature and the base of the fin. As far as I can see this sheath of connective tissue is chiefly built up by a large number of these lamellae, but they are so inextri- cably united that I have not been able to follow lamella 6 in this sheath. If starting from the fin, we prepare free one muscular ele- ment of the fin, and this element is lifted up with enough precau- tion, it may be seen, that from the base of such a fin-element as well a thin lamella of connective tissue passes into that sheath of tissue in which we could follow the reversed part of lamella 6. The direct continuity however of both lamellae in the sheath of dense connec- tive tissue, I have not been able to establish. The muscular elements composing the fin (figure VI) are trian- gular laminae; one side of the triangle is contiguous to the fin-rays and the connective tissue which unites these rays in the mesial plane of the body, the lateral side forms part of the lateral surface of the fin, while the base is excavated and moulded upon the shallow depression in the dorsal musculature. From the outside a septum of intermyotomal tissue (s.i. figure VI) penetrates into the muscular substance of the fin dividing this substance into a lateral (6) and a mesial part (a). This septum inserts a little above the muscular substance upon the fin-rays, and becomes thinner and thinner without ( 817 ) reaching the base of the fin. At the base therefore the lateral and mesial parts of the muscular substance are continuous and form a peak (figure VI p. 1), lying quite near the mesial septum of the body. This peak must therefore represent the peak which is lost in the transverse section (figure IV) made at the level of the first dorsal fin. The septum penetrating into the muscular substance of the fin is therefore the intermyotomal septum stretched out between lamellae a and + of figure II. It ensues therefore from the combined observations, that the first dorsal fin (and the same applies to the other dorsal fins) is differen- tiated by a process of infolding similar to the differentiation between the dorsal and the lateral and between the lateral and the ventral mus- culature. The line of infolding crosses lamella 6. In that part of the lamella, which lies in the depth of the fold the muscular tissue is atrophic. Proceeding from peak 2 caudally along lamella 5 the atrophy of the musclefibres gradually increases, whilst on the other hand proceeding from peak 1 caudally along lamella 5 (as far as it lies upon the fin) the atrophy of the muscular tissue is abrupt. The position of peak 1 has not changed in respect to the mesial plane of the body, only the lamellae have changed their direction. The superior cornu (i. e. lamella a) is no longer directed cranially but turned upwards and this is also the case with that part of lamella 6 that has passed into the fin. In connection with this representation of the facts, I deter- mined the direction of the muscle fibres in the fin; here they slope downwards from the intermyotomal septum. Now if we imagine the lamellae composing the fin restored to their original position, the course of the fibres in lamella « would be from mesial to lateral and from caudal to cranial and this was actually the direction. of the muscle fibres in lamella qa in the region cranially of the first dorsal fin. Differentiation of the latero-ventral musculature. The lateral musculature, as described by vaN BisseLick, shows a peak directed caudally (peak 5, figure Hand fig. VII) situated near the second line of infolding L'L'. Proceeding along the body a second peak appears directed cranially. The first myotome showing this peak (peak 6, fig. II and fig. VII), is the eleventh myotome following the first myotome giving a muscular element to the first dorsal fin. The two peaks lie near to each other in the neighbourhood of the second fold. In consequence of the infolding of the myotome at that place, they do not reach the surface of the body, being covered from the outside by the ventral musculature. Meanwhile the ventral part of the myotome undergoes a change in form, the first lamella belonging 56* ( 818 ) to the ventral musculature (lam. f figure II and fig. VID) becomes shorter and the first peak of the ventral musculature directed crani- ally (peak 7, figure II), more and more develops into a true peak. Now by the disappearence of lamella f peak 6 and 7 approach each other, remaining divided, however, by a thin lamella of connective tissue penetrating into the second fold alqng the line LL (fig. VIII and IX). In consequence of the process of infolding peak 6 lies mesially in respect to peak 7 which covers peak 6 from the outside. At the level of myotome 15 (reckoned from the first myotome, giving an muscular element to the first dorsal fin) the second fold vanishes. Together with the disappearence of the fold we notice the vanishing of the displacement of the lateral musculature in respect to the ventral musculature, which was only a consequence of the process of infolding, so that the two peaks (6 and 7) lie side by side in the same transversal level of the body. At the place of disappearance of the second fold the two peaks unite to a single peak directed cranially. Together with the disappearance of lamella f we notice the further development of lamella g. At the same level where the second lateral fold disappears, we find the appearance of the cartilagineous plate, uniting the two basipterygii of the pelvic fins. With its front border, this plate folds in lamella g (figure IL and X) from the inside so that this lamella covers the front edge of this plate; in this way the pelvic fin is formed. The details of the formation of the pelvie fins I have not yet investigated. By the formation of the pelvic fin peak 8 (fig. II) and Jamellae g and 4 pass into the musculature of the fin, so that in a transverse section through the animal, at the level of the pelvic fin, the trunkmusculature is only composed by five peaks (viz. 2. 3. 4. 5. (647) of figure II). -This structure of the transverse sections does not change any more proceeding along the body caudally (figure XI and XII). The disappearance of the first fold, dividing the dorsal from the lateral musculature, takes place in the same way as described for the second fold, at the level of myotome 45 (reckoned from the first myotome giving an element to the first dorsal fin); only the case is more simple not being complicated by the presence of two peaks. Finally I paid attention to the influence of the abdominal cavity upon the form of the myotome. I found this influence to be very restricted, as it only determines to some extent the dimensions of the myotome, without producing any particular differentiation in its form. In fine I wish to express my thank to Prof. LANGELAAN for his aid and assistance in these researches. Fig. Il. B, VAN TRICHT. On the influence of the fins upon the form of the trunk-myotome. Vol. IX Proceedings Moyal Acad, Amsterdam ( 819 ) Anatomy. — “Anatomical Research about cerebellar connections.” (Third communication). By Dr. L. J. J. Muskens (Communi- cated by Prof. C. WiINKLeR). (Communicated in the meeting of March 30, 1907). The ventral cerebello-thalamic bundle. Whereas it is nowadays generally aecepted, that the direction of conduction in the superior crus cerebelli is cerebellofugal, there is no uniformity of opinion attained yet by the authors regarding the bundle, which is found degenerated in the predorsal region in the pons after cerebellar lesions. After Prrrizzi and van GEHUCHTEN, THOMAS, ORESTANO, CAYAL and Lewanpowsky this bundle is built up by fibres, which take their origin from the superior crus after it has crossed the raphe in WeERNEKINCK’s commissure, the direction of conduction being rubro-fugal. Prosst however, describes this bundle as the ventral cerebello-thalamic bundle, conducting nervous impulses from the cerebellar basal nuclei upward towards the red nucleus. The problem of this bundle really has to deal with two questions ; 1st. which are the two nerve centres, which are connected by means of this bundle and 2"¢. what is the direction of conduction of impulses in the same. Cats appear to be more suitable for these experiments. In two animals different parts of the cerebellar cortex, with the adjacent part of the basal nuclei, were removed, except the flocculus. In these animals there was hardly any degeneration at all in the ventral cerebello-thalamic bundle, whereas in 3 other cats in which with other parts also the flocculus was removed there was very extensive degeneration of this bundle. That these fibres do not take their origin from the cortical gray matter of the floeculus is proved by the fact, that in another cat in which the cortex of the formatio vermicu- lavis cerebelli was injured, no degeneration of the said bundle was found. In two cats (XXIII and LXI) a lesion was effected in the mid- brain, by passing a curved knife in front of the lobus simplex cerebelli in such a way, that the predorsal region on the right side was cut, distally from the red nucleus. In none of these animals any degeneration was found in the ventral bundle. If Cavar’s sup- position were correct, certainly a great many of the descending col- laterals of the superior crus ought to have been found degenerated. In one cat (LVI) a longitudinal lesion was effected in the teg- ( 820 ) mentum, the instrument (Prosst’s covered hook) passing through the middle crus cerebelli. In this cat were found a certain number of fibres degenerated, which passed through the regio reticularis of the side of the lesion and then, crossing the raphe and running upward in the predorsal region of the other side, took their way towards the red nucleus. This experiment tends to show, that there are direct fibres, coming from the basal cerebellar nuclei, which do not join the superior crus, but follow the ventral course to arrive at the red nucleus. Lewanpowsky,s fibres O. P. (in fig. 66 and 37) are not to be identified with these fibres on account of their entirely different course. In cat LXII the anterior crus cerebelli was partially cut, and at the same time an incision made into the middle crus. Also in this animal there was found no degeneration on the distal side of the lesion except the bundle of Monakow. Were the ventral bundle to be regarded as being formed by descending collaterals of the anterior crus this result could hardly be explained. A simular result was obtained in cat LXVILI, where hemisection of the pons was effected. Also here there was no degeneration on the distal side of the lesion. Mathematics. — ‘“Lquilibrium of systems of forces and rotations in Sp,” By Dr. 5. L. van Oss. (Communicated by Prof. P. H. SCHOUTE). (Communicated in the meeting of March 30, 1907). Referring to the following well-known properties : a. The coordinates pij and arj of a line p and a plane a satisfy the five relations: P; — Pl Pjm == Ply Pkm ae Pjk Pin — Qo. a = Thiel Lim = 0, . (1) t of which relations three are mutually independent. 6. The condition that a line p and a plane a intersect each other is expressed by = pij Tij == 0 . . . . ° . . . . (2) c. The coordinates of the point of intersection X of two planes x, x' and that of Sp, through two lines p, p’ are: Ty == Th Ie jm aS HIj Kam + TT iI: I In == JT jm A kl + km Rij + Him Te jk he = Pil P jms acy 10, EN vd od anaes (3) a Se we wish to draw the attention to the following properties : If (7) are ten arbitrary quantities: i.e. not satisfying the relations Geni) = (jk) (lm) = 0, we shall continually be able to break up each of these t quantities into two parts (4) and (7)", so that (4) (jm)'= (kl)! (jm)"=0. It is easy to see that this decomposition can be done in oo* ways. For each decomposition holds good : ; sana (ja) == ae (hl) (fin), tee EN for : = (kl). (jm)! = = (Al)! {( jm) — (jm)} = = (Al)' (jm), likewise : > (Hy (jmy" = (U) (jm), from which by addition : > (UI)! jm)" = > (Kl) (jm). Giving a geometrical interpretation we regard a homogeneous system of 10 arbitrary quantities aj and aj as the coordinates of a system a of c* lines, in pairs a system a of oo* planes, in pairs ' a, a conjugated by the relations: a', a" conjugated by the relations: Oije ienke (5) dij + aly = dj (8) All these lines lie in one Sp, = All these planes pass through having as coordinates: one point X, having as coordinates: Si = > Akl Aim ome verts (6) U, == Pe Akl ajm- Ce ar (6') c 4 We now annul the homogeneousness of the p-, x-, a- and a-co- ordinates. This causes those elements to assume vector-nature and makes them interpretable respectively as force, as rotation, as dynam and as double-rotation. The equations (5), (5’) determine the reduction of the vectors a and e@ on the conjugate pairs of lines and of planes of the systems a and « under consideration and not yet partaking vector-nature, whose structure now becomes revealed. Il. In connection with the meaning given in 6 of the equation = pij ij = 0 we interpret gn a) a, TELA RIEN as the condition that a linepcuts as the condition that a plane x a pair of conjugated planes of cuts a pair of conjugated lines system a. of system a. This gives us a very fair survey of the structure of the linear complex of lines and planes. The reduction of the equation of the complex of planes to its diametral space is now easy to do; likewise ( 822 ) the further reduction to the simplest form (47) = 6 (jm), assumed by the equation when the edges #/ and jm, the planes jm and chi of the simplex of coordinates are conjugated elements of the systems a OF &. Ill. If we assign to the elements p, a, a, @ vector-nature, expres- sions aj pj, Saya become of importance as virtual coeficients (in Barr’s theory of screws) and the disappearing of these coefficients then gives the condition that the force p performs no work at a displacement in consequence of a double rotation a, resp. that the dynam a performs no work at a rotation a. So in Barr’s notation the equations (7), (7) give the condition of reciprocity between force and double rotation, resp. between dynam and rotation. In like manner the equation DN SO eta Nn etc ane (8) which includes (7) and (7)’ and likewise (2), gives the condition of reciprocity between the dynam a and the double rotation a. IV. We shall now pass to the general equilibrium of forces and rotations. It will be convenient to understand by p, zr, a, a vectors unity and to indicate the intensity of these vectors by a factor. It will be sufficient to limit ourselves to the equilibrium of forces, leaving the treatment of the dual case to the reader. In the first place we regard the case of ” forces, n > 10 working along lines given arbitrarily. It goes without saying that for the equilibrium it is necessary and sufficient that the intensities /” satisfy the ten conditions: 2 APO tr AEEA ee Oe ee We can therefore in general bring arbitrary intensities along 2 — 10 vectors, those on the other ten then being determined by the above equation (9). In particular for n= 11 the theorem holds: To vectors along eleven lines given arbitrarily belongs in general only one distribution of ratios of intensity, so that the system on those lines is in equilibrium. The generality of the case is circumscribed by the requirement that uo ten lines ean satisfy one and the same linear condition in the form Se gee 0, where the coefficients a, do not depend on Dj bi in consequence of a well-known property of determinants tending to zero. ( 823 ) So if there are among # lines at most 10 belonging to a linear complex we can satisfy the equations (9) by choosing all intensities except those belonging to these 10 equal to 0 and then (if not all subdeterminants of order 9 tend to 0) we shall be able to bring along these last only one distribution of intensity differing from O in such a way that the system of forces obtained in this manner is in equilibrium. We have thus at the same time arrived at the following theorems: For the equilibrium of ten forces it is necessary that these belong to one and at most to one linear complex. In this case always one and not more than one distribution of intensity is possible. If we continue the investigation of the equations (9) we then obtain successively the conditions of equilibrium of 9, 8, 7, 6, 5 forces. We can express the result as follows: In order to let n forces, 11 >n>4, admit only of one distri- bution of mtensity in equilibrium, it is necessary and sufficient for them to be the common elements of exactly 21—n linear complexes. In particular for 2 = 5 we find the condition that the forces must belong to a system of associated lines of SEGRE. This has given us a connection with a former paper in which we treated this case synthetically. V. The condition that ten forces in equilibrium belong to one complex follows almost immediately out of the interpretation of the equation Lay pij = 0 as condition of reciprocity of force and double rotation. Let e.g. ten forces be given in equilibrium; nine of these forces chosen arbitrarily determine a complex, so also the double rotation « for which none of them can perform labour. The united system of ten forces, as being in equilibrium doing no labour for no motion whatever, it is necessary for the tenth force to be likewise reciprocal with respect to the double rotation a, ie. this force belongs with the former nine to the selfsame complex. Equally simple is the deduction of the conditions of equilibrium for nine forces. For eight forces determine a simply-infinite pencil of complexes whose conjugate double rotations a + da’ are all reciprocal with respect to these eight forces. So they must also be reciprocal with respect to the ninth force in equilibrium with these, i. o. w. the latter must belong to all linear complexes to which the eight others belong. And so on. ( 824 ) VI. We shall now denote still, by means of a few words, in which way we can arrive at an extension of the screw-theory of BALL by the application of the principle of exchange of space-element to the 10 equations 2 a;§;= 0. 1 By interpreting this equation either 1st. as condition of united position of a point X and an Sp, Zin Sp,, 2.4. as condition of reciprocity (Barr) of a dynam X and a double rotation Z, we make a connection between the point- and Sp,-geometry in Sp, on one hand and the geometry of dynams and double rotations on the other hand. To each theorem of the former corresponds a theorem of the latter geometry. Nov the remarkable fact makes its appearance that the fundamental theorems of the geometry of Sp, correspond to the fundamental theorems of the theory of screws of Batu in Sp, With this as basis we shall show, though it be but by means of some few examples of a fundamental nature, that the principles of a generalisation of the theory of screws are very easy to be arrived at by transcription of the simplest properties of the point- and Sp,-geometry in Sp, which examples can at the same time be of service to explain the above observations on the theory of BALL in Sp,. To avoid prolixity we introduce the following notation. We call: dynamoid the system of lines whose conjugate pairs can serve as bearers of a dynam. rotoid the system of planes whose conjugate pairs can serve as bearers of a double rotation: So dynamoid and rotoid correspond to dynam and double rotation as in the notation of Barr “screw” to dynam and helicoidal movement. Let the following transcriptions be sufficient to explain the appli- cation of the above principle. oX: Point X bearing a mass Dynamoid X bearing a dynam 0. of intensity X. oF: Sp, = with a density of Rotoid 3 bearing a double mass 0. rotation of intensity 6. (X'X"): Right line, locus of the Pencil of dynamoids, locus of centres of gravity of the bearers of the resultants of variable masses in the two variable dynams on the dyna- points X' and A. Sps-pencil. A right line has always moids X' and X”". Pencil of Rotoids. A pencil of dynamoids always ( 825 ) a point in common with an Sp,- An Sp, is determined by nine points. p spaces Sp, cut each other according to Spo—p. Etc. etc. contains a dynamoid reciprocal to a given rotoid. A rotoid can always be deter- mined lying reciprocal with re- speet to nine dynamoids. The dynamoids reciprocal to the movements of a body with p degrees of freedom (9-p)-fold infinite pencil. form a We shall now apply the above to the problem: “To decompose a dynam according to fen given dynamoids”, this problem being a tran- scription of the following: “To apply to ten given points a distribution of mass so that the centre of gravity finds its place in a given point.” We again put side by side the results. To be defined successively : a. An Sp, through nine of the given points. 6. The right line through the remaining point and the centre of gravity. c. The point of intersection of this right line with the Sp, found il ia. d. The decomposition of the mass in the centre of gravity according to this point of intersection and the 10% point named in 5, which is possible, these three points being collinear; gives at once the mass to be applied in the last named point. The other must necessarily be- come the centre of gravity of the remaining nine points. e. These treatments to be repeated for the determination of mass in the other points. Zalt-Bommel, March 28, 1907. The rotoid reciprocal to nine of the given dynamoids. The pencil of dynamoids through the remaining dynamoid and the bearer of the given dynam. The dynamoid on this pencil reciprocal with respect to rotoid found in a. The decomposition of the given dynam according to the dyna- moid found in ¢ and the 10% dynamoid named in 6, which is possible, these three dynamoids belonging to one at once the dynam on dynamoid. The other must necessarily bear the resultant of the dynam to be applied to the remaining nine dynamoids. These treatments to be repeated for the determination of intensity on the other dynamoids. the pencil, gives intensity of the the last mentioned ( 826 ) Physics. — “Contribution to the theory of binary mixtures,’ UI, by Prof. J..D. vaN DER WAALS. Continued, see page 727. We shall now proceed to describe the course of the spinodal curve and the place of the plaitpoints when choosing regions of fig. 1 which lie more to the right. But it has appeared from what precedes that to decide what different cases may occur, we must ae dp dp know the relative position of the curves —— =O and mik ao da? dv? : dp Ek : ae which now the curve —---=— 0 is added; so the relative position at ardv different temperatures of the three curves which occur in the equation of the spinodal curve. 3 42 aw : ent = 0 and ——~- == 0 may be considered as sufficiently av arav b] The curves. known, and the knowledge of the relative position of these curves with regard to each other enabled us already before to elucidate sufficiently the critical phenomena of mixtures with minimum critical temperature and though with regard to the relative position of these lines some particularities are met with, which have not expressly been set forth, I shall assume the properties of these lines to be 42 d w , i known. But the eurve —- = 0 is less known — and it has appeared at from the foregoing remarks, that if we wish to understand the oecur- rence of complex plaits, the relative position of this curve with dp EE zi == 0 Mist De known. If this line lies alto- dw ra to speak of on the course of the spinodal line, but if it les either partially or entirely outside this region, the influence on the course of the spinodal curve is great, and the existence of this curve accounts for the complexity of the plait and gives rise to the phenomena of non-miscibility. I have, therefore, thought it advisable to investigate respect to the curve gether within the region where is negative, it has no influence 2 d : the properties of the curve on A 0, before proceeding to the descrip- : da” tion of the course of the spinodal curve also in other regions of fig. 1. A perfectly exact investigation of this line would, of course, require a perfectly accurate knowledge of the equation of state. But ( 83% ) the value assumed already before as an approximate equation of this line: (=) da th 1 de dar? ENE Wee <6 ee an | dx” ede) (vw—by v will prove adapted to give an insight into the different possible are : ; : : wy dp positions of this line with respect to —- == 0 and —— — 0. dv? dadv d? THE CURVE — dx? The differential equation of this curve: dp NG “dw dp da - dv LE =S 0 da’ fea ear may also be written in the following forms: ly dw Ty dT dx +* yy — —__ —__— 0) UT datig or dw : dw ; a? (e—w) dT 0 le -+- - lv — = da? En der TT or aw dw de dT SS dt lv — NE dal Sok or 20 1 dw ; lL ead? 0 We - v — — -——_ = dx? dx*dv v dz? T My oe The curve cate can only be found for positive value of 7’ AL ( dake. ee 2. ; 5 : when — is positive. So we derive from the latter form that da? I 3 de É ’ rise 8 d $ : (F) is positive for the points for which + is negative, and the a 5 dav’ other way about. dv ‘ toe ; In the same way, that (=) is positive for the points for which a T dw ‘ ; 4 = Ni . : ; ees is negative and vice versa. The transition of the points for Ad : (28280) dw . Ae : which aan oe negative or positive, takes place in the points of the ar aw : i ae : curve mr — with maximum or minimum volume or for which A& dp dw dp — == 0; and the transition of the points for which — = — —~ da* P duda? dx?» P is negative or positive, takes place in the points with maximum or or minimum value of «. From all this follows that the curve dp Te contracts with rise of 7, and has contracted to a single Ak point for certain value of 7’= 7;,. It is now necessary for our purpose to determine the value of Jin and also the value w, and v, of the point at which this locus vanishes. This means analytically that we have to determine the values of 7, w and v, which satisfy: dw dw Mw dp au == 0, = : — and i! == ae = dix? de dvd? do? or the equations: (1e) (v—by? 6) db? Pa del 1 da? and MRI (By aE Ree (3) If A) is divided by (3), we get a relation between « and », which in connection with (2) may lead to the knowledge of a, and vg. Then we get: (v—by? a an -=» + b= —6) + 2b a(1 (5) - at lb \? 227(1—2)? and as (v-—b)? = ) gie) we find: da 1-2 b e(l—z) 1 (2e (l—e) je p dT Tom Bel de (4) ( 829 ) lb and putting b=6b,+ 2 me da b jee id 1 | 207(1 yr Iz 5) 1 fan fede 1— 20 \ de The Ist member of this last equation representing the ratio between the size of the molecules of the first component and the difference of the sizes of the two kinds of molecules, we see that «, depends only on the ratio between the sizes of the molecules of the two components. If we take the two extreme cases 1*t that 5, may be put equal to 0, 2rd that 6, is equal to 5, we find the two extreme values of vy. Et (1 90)" ox Ba? Sa Zar SS as = IE i 0: it Me oe Ee & = “1 ane Papi a A): OR ae ee “ db or «='/, for b,=0. For the other extreme case ge we find & ii pe For some arbitrarily chosen values of 2, I have calculated the 1 corresponding values of eel b, v—b Ly ann 5 Yq (see p. 832) ae eene 0 St aa sf NE OS 0,4 nne Oe Ue A De AK OE 12S ere (as ati oe OD hoe le aoe 3.08 sae ce ORT eae ae IR Ue Sees O06 OS a OSes, he ee dered ape vu. 5204 ob |) oe GaGa a> ea eee Sete aie ., LO, OL ones ela Oa ote eae en SN 0,5 eet ttt 0 ros AND Re le Wl 8 If on the other hand the value of #, has been caleulated by the b aid of the given value of En v, is determined by the aid of the Jb de equation : db (3 /2r(l—x)' OT a EEC ay db f If --=0, in which case a, = '/,, this equation gives an indefinite ( 830 ) value for v— bh. So it is better to express v —b in a value in db which aa does not occur. We write by the aid of formula (4): rs Pe 2x (1—e)' 1-22 ; xv(1— x) 1 Va 223(1—.)? er ee 1—2e v—b 2 b Bee 2 (12)? v—b In the above table we find the vadues of in calculated for arbi- v —b)=b or trarily chosen values of a,. For values of z, differing very little v—b approaches 2B~ (1 — 2a,)?. from '/, the value of b The value of MRT, may be brought under the following form : da (1 — 2a)? | as i Cay pase as ON, Ar(l— 2) MRT, Erle en b 12: (1—2z2)? |! )? 4a(1—z) |” dy ; So the temperature at which the locus PE =— 0 vanishes depends AL in the first place on the value of « at which it vanishes, and in Ma 3 5 the second place on the quantity ie As according to form. 5 & da da 1 1 de? may lie between = and a? the factor of ee Tay YAU between EL ) 1 and re The value of that factor is therefore only determined by the E EUD ratio between 6, and 6,. For 5, —0 the value is Sik for b, = | this value is rh So the greater the difference in the size of the molecules, the lower this factor, and the lower the temperature at 2 dp which el = 0 has disappeared. And because the non-miscibility in ar d? the liquid state is to a great extent due to the existence En in U ( 831 ) Ma ; molecules of the same size & being always thought equal) will el not so readily mix as those for which the size of one kind greatly exceeds the size of the other kind, a property to which we might have concluded without calculations. But in the second place the 3 id ; , quantity Pe 2 (a, + a, — 2a,,) has great influence on the height of „2 ; ee this temperature, and indeed, -in so high a degree that if re should av 2 b) Be =="0, the locus = 0 would already have disappeared at the Av absolute zero point. Indeed, we might have seen from the very Ma beginning that this locus could never exist for Pie negative. Every- thing, on the other hand that diminishes a,,, makes 7, rise, and so furthers non-mixing. In some limiting cases we may compare the value of 7, with that of 7%, 1** in the case that into a given sub- stance we should press a gas as 2rd component following the laws of Borre and Gay-Lussac perfectly. For such a gas we should have to put 6, and a, equal to zero, and so probably also a,,. The value 1 of w of the formula for 7, is then equal to 5° The a for the mixtures ee MPa containing only one term then, and being equal to «,z?, a Saag adt The value of 5, for the mixture is then equal to 4,7. On these suppositions MRT, = 8 a, pas ar — —, and so 7, is equal to the critical tem- 27 b, perature of the 2ed component. The value of 7). for every mixture taken as homogeneous, is then equal to «(7%.),. Consequently a I } T, =3(T;),. For a value of 7’ somewhat below (7), the locus Py dv? =O is restricted to a very narrow region on the side of the 2 Wo ~ = 0 still exists, and may be compared to Ak 2ed component, while a small circular figure whose centre is a point with the coordinates x='/, and v=b,. The spinodal curve has then an equation which may be written as follows: MRT 2a Proceedings Royal Acad, Amsterdam. Vol, IX, ( 832 ) which equation shows that it consists of two straight lines, which join the point z=0O and v=O with the points for whieh —“" —0 for the second component. At temperatures which are not too far 12 d : as = 0 lies, therefore, entirely outside the & below (7%),, the locus 2 d : : EN ; = curve = 0, and is then restricted to the left side of the figure. v 2nd, As second limiting case we take 6, = i but a, differing from a, Pax(1—z) 2(a,+a,— 2a,,) Then MET, = cage GEK and because # = — > MRT, = 15 —, 8 Mien whereas MRT). is equal to : etl 57 1 for 2 '/,. ‘Then; aos: T, may be larger than 7), viz. when — (a, +a,—2a,,) >(a, +a, +2a,,) 23 or if 2a,,< = (a a,). But even if 7, should be < 7%, this implies ieee a ° ; dap by no means that shortly before its disappearance the locus = 0) av ule te WENT 0 : lies in the region in which zn 8 negative. The previously calculated av v—b values of rea show that this disappearance takes place at a very small volume, which may be smaller, and in the limiting case will certainly 2 d be smaller than the liquid volumes of the curve —_ 0. To ascer- av J2a0, d?yp tain whether the disappearance of = 0 takes place in the region at 73 EF d in which \ dv is negative, we may substitute the value of 7, x and dy = ; vg in the form for ——, and examine the conditions on which this 5 vy? je valne of heepmes negative. If we write for Mr, dv? 4r(l —ary) == yy", then URT, — @atg(l1—azg) l—yg vg Dg __ 2y, Ug 1449 4 dat Bg Ad vnd Pe and (Ee) ENG g rE Aue (he DLS ( 833 ) Py So the sign of (| i dv? 2a (vg—by)? bg or on ) depends on MRT, — 7 Ge B RC da x(1—a,) 1—y, Za 4y7 1—y, dix? by = (1+ yy)? fs by (L+-yq)? 14-90 dy, Ly For the discussion of this last quantity we first put the first mentioned limiting case, in which a, and a,, may be neglected with or on (a, + a, — 2a,,) x, (1 — ay) —a 1 respect to a,, and a == a,v* may be put; the value of y, being = 5 according to the table of p. 829. With this value this quantity becomes: Pe Lengel 3 Asa so positive. For the other limiting case for which y, = 0, it is also positive. But for the intermediate cases, specially those for which a, + a,—a,, is small with respect to a, and 6, and 4, are not equal, it will be . ; : 8 dp negative, and shortly before its disappearance the locus nn 0 dw will lie in the region in which DE, is negative, and the existence of v this locus will have little influence on the course of the spinodal curve, and accordingly it will not give rise to a complex plait *) or rather to a spinodal curve which diverges greatly from the curve d? Zi dv? Ship NE Let us now also examine where the point in which en ==) ar ; F > dy dp disappears, lies with respect to the curve = 0. or to | — |= arav Let us substitute the value of MRT, x, and v, in the expression for (2 . If this expression becomes positive, the point lies outside av the curve or rather at smaller volumes than those of the curve d (2) = 0 and the other way about. Then we find for: & v 1) I need hardly state expressly, that in this communication I no longer attribute the cause of the complexity of the plaits exclusively to the abnormal behaviour of the components, to which at one time I thought I had to ascribe it, in com pliance with Leuretp. On the other hand it would be going too far to deny the abnormality of the components any influence. 57* ( 834 ) db da da da wR? Se (v,—b)? v? da da a(1—a) 1—yg db 1 de de? 6 ‘(1+-y4)?da(vg—b)? vw’ b = ag(1—a PR and after the substitution of — = ral tn and of 9 = de 4 e 15 (1++-y,)? the sign proves to depend on the expression: da da : Ze (1—2z,) — = 4y,’. lr : : de da 1 In the first limiting case in which Ee 2a, , ae 2a,%g , 99 == 1 and a = 3° this expression — 0. Also in the second limiting case, 1 ieee in which ome and v,=0. So in the limiting cases the curve Py ; dap — 0 intersects the curve dadv de: the latter disappears. Also in an intermediate case this quantity may be zero, but the value of z,, at which this takes place, depends on da up to the last moment in which da? (a,—a,,) — (4,,—4,) — or on A da (4,,—4,) (La) -+ ay (a,—4,,) da MPa a,—a dc? A [fi we (write then = =: and, ay da 1—ag+<a,(1+ A) dx A at . —-———. We then derive the value of az, for this intermediate 14z,A ; case from the equation: (1—22,)*/s [429( 1— &y)| 23 Fak en 12, Vs 1+Aa, [162,(1—e)? | —_—_— (1 — 2a,) = 4 TELEN 9) or ( 835 ) For values of a, differing little from } we find approximately: 1 athene A gr Tre If for A we take the value 4, which must be considered large for 1 1 molecules of about the same size, then aye would be = zz: The conclusion which we draw from all this as to the situation of the 2 ae Me point in which = da? dy de dv OQ disappears, with respect to the curve = 0, is the following. In most cases this point disappears within dp ; : dp\ , : the curve ( = 0, and so in the region where { — | is negative, de), de), d, but this can also take place on the other side of (2) = 0, so at “a/v a volume which is smaller than that of this curve. a,+a,—2a,, That at positive value of A, so at positive value of A eg (A2) 1ltaA ella) we represent the two members of this equation graphically. The first member, namely, represents then a branch of a hyperbola which at 2—O has a height above the axis of w equal to that of 4, and has always a root, appears immediately when A 8 at «—=1 a height equal to TA and which, therefore, proceeds continuously at a certain positive though deereasing height above the x-axis. The second member represents a line which for «=O has a point infinitely far above, and for «=1 a point infinitely far 1 below the «z-axis. This line passes through the point «= a and on the left and the right of this point the ordinates are equal, but with reversed sign. So intersection will certainly take place, and for d 730 disappears 1 positive 4 at a value of ore For the case that d at smaller volume than that of the curve 0, the first member i must be larger than the second. As A is larger, the point of inter- ; 1 : section will be further removed from 5 and so the series of the values of z for which the condition is that the first member be larger ( 836 ) than the second member, has increased. From this we conelude that Py : : ; 5 : ——=() may disappear also for very different size of the molecules dx? : dp ae Ee in the region for a positive, if A has a considerable size. But for ar ; 1 perfectly unequal size of the molecules oo mere would be 1 1 3 >> 3 = > 1, which is not yet satisfied t A DANO) ype ‚ which IS not yet Satished even a == 0: OES ae ey a N . ‚(dp dep. Fig. 6, in which the intersection of | — )=0 and — ==0 has v at at been drawn in both points on the left of the point in which dp rte : : GA = 0 has the minimum volume, holds for this latter case. The [2 point in which an Q disappears, must viz., lie on the line U 3 d*p 5 -=—0. As has already been mentioned before, this line passes at av dp ' through the point where (2) = 0 has its smallest volume, and as ; dv is easy to calculate — is then always positive. If now in fig. 6 ae 2 dr : \ d*p the line ae = 0 contracts, and it must vanish on — —0O, then the at at point in which it disappears, lies at smaller volume than that of dp : Ee Ee) — 0. For the opposite case the two points of intersection must CO therefore be drawn right of the point with minimum volume. Also the intermediate ease has now become clear. In this respect there 2 is an inaccuracy in the drawing of fig. 5. The line Pele. which Md has already almost quite contracted, must be expeeted there on the C right of the point in which (2) — 0 has the smallest volume. So AL v the line (3) =O would have its minimum volume more to the da); left in fig. 5. In fact, with rise of temperature all these lines are subjected to displacements — however, not to such a degree that the relative position is much changed by it. ( 837 ) All these remarks seem essential to me for the following reason: we shall, namely, soon have to draw the relative position of the dp dp curves REE O and —-==0, also in regions lying more to the right mh U of fig. 1, in order to decide about the more or less complexity of the plaits at the different temperatures. Then we shall have to make assumptions as to this relative position, which otherwise might seem quite unjustifiable. A great many more similar questions would even have to be put and solved, before alle doubt as to the validity of the assumptions would have been removed. And it remains the question if for the present the imperfect knowledge of the equation of state for small volumes does not prevent our ascertaining with perfect certainty whether a phenomenon of mixing or non-mixing is either normal or abnormal. So, before proceeding to the appli- cations I shall subject only one more point to a closer investigation, viz. the question whether in the critical point of a mixture taken 2 rs OD Pye as homogeneous, the quantity aoe positive or negative, so the at db\? _d?a 8 a da da? 1 27 6 jp (1e) 4b? sign of the quantity: 36 or of ; db\? da 1 je 9 da? a aoa Abre A Ban d'a da’? : : As 2a saat a + 4 (a,a,—a,,"), we may also write for the wv wv last form: Gl ey AS An 2 i! | da) 9 \de a 9 a, a,—a,, wv (1— 2) Ab? 16 a* 4 a? As a first special case we consider a substance mixed with a periect “gas then >, = 0, 4, = 0 and a, 0. Hence, aaan 6—b,«. With these values the above form becomes: i 2 3a—2 e(l—e) 2 «2? (1—2) SO — is negative in the critical point for values of «<< °/,; for d? dl, the curve e= will pass through the critical point. But x ( 838 ) for « > 7?/, the two curves will lie outside each other, as has been drawn in fig. 10, and already observed above. For all other cases in which a and 5 cannot be equal to zero, the value of the expression for «=O and «=1 will be positive infinite. If it can become negative, this will, accordingly, have to be the case for two values of z. Now very different relations may exist da 1 db between — — and — —. Thus 1 da 1 db a. —— =*/;— — for the plaitpam a der b da Fig. 10. circumstances of a mixture taken as homogeneous. ') With these values the form reduces to a quantity which will certainly be positive, as even if a, a, > a,,’, the value of a, 0. ais + 1 at : - can probably never be larger than — — — —, the mini- a” : 9 z(l—e) 16 mum value of which is 1 db 1 db Rae eRe the sign of the form under discussion, depends on the ) aL a dt da Idd, » 1 ab 16 Tele ‘ ; value of AE If 5 pe bs B a negative value of the form is possible. So for mixtures, AE in which the components differ greatly in the size of the mole- den Re ee cules, the case of fig. 11 occurs for srl minimum 7%, and this minimum 5 value of 7 could not be rea- lised. For mixtures, for which 1 da 1db fl dby ( ) may be ne- - -- a dx~ b da’\b de 4\a du which is even perfectly allowable in the limiting case, for which dp 2 . 9/1day glected with respect to ae : b-==;, and will be negative, Rig e 1a da In all the above calculations the equation of state has been applied with ( 839 ) 8 (1 da 1 oh pel 4 \a du vel —a) As minimum value for which this is the case, we should then have: 1 da 8 when we put: In all such eases, in which the critical circumstances of a mixture 2 en Een … ay taken as homogeneous, fall in the region in which — < 0, these ar circumstances are not to be realised. Nor are they to be realised when dw da? distance round this point, and the plaitpoint circumstances are not very different from these which would be the critical circumstances 2 with an homogeneous substance. If Ten < 0, a considerable difference v > 0, but then the spinodal curve passes at least at a small may be expected. ‘y. k THE SPINODAL CURVE AND THE PLAITPOINTS WHEN —— IS POSITIVE. at Let us now again proceed to the discussion of the course of the spinodal curve and the plaitpoints; but now in the case that with increasing value of 5 the quantity 7) rises. Let 77%, be much higher than 7%, , J2 a. a ree ‚dy and an Now two cases are possible. The value of — may be ar positive or negative in the critical points of every arbitrary mixture. 2 ) For es ==0, and in general for very small value of «, where — e(l —«) dp. ae j is very large, rib. certainly positive, however large the value of ( Hv da WEE —: a may be, and also for values of « differing little from 1. da:* : da dx? 2 ' aw. oe : ips ‘ : , is small, ae is positive in the critical points of all mixtures. a aL If 2 » But for large values of — : a there are two values of x, between AL value of 5 not depending on v. Hence in this equation we get the factor */, for 4 : 5 which, as we have already repeatedly observed, we should really substitute —. 6 ( 840 ) dp Mts 5. et which — is negative in the critical points. If in this case we draw ( a _ ay iene ds —0( with a top, either at «=O, or at a small value av [2 of z, the curve —- = 0, which is chiefly restricted to the left side aL iw - = 0), andgon the of the v,«-figure, lies partly outside the curve ; v small volumes. If we now apply the reasoning of p. 737 etc. also J2 in this case, when we had the reversed state as far as == (Kas av concerned, we conclude that for large values of v the spinodal curve dp NEL does not move for away from —— =O, but that it is forced back av ; dep . ; to smaller. volumes for those values of «, where —— is negative, Av 4 Py J : and draws again very near to ay 0 with very small values of 2. av Naturally the course of the g-lines in connection with the course of the p-lines must indicate this. The course of the p-lines for this case must be derived from the right side of fig. 1, from which appears at the same time that the ad? : curve ——=0O occurs, but with avtav sensibly smaller volumes than 2 those of el = 0. And the course adv” of the q-lines is then indicated by fig. 5 or perhaps sometimes by fig. 6. In fig. 12 a couple of p- lines have been drawn and a q- line which touches these p-lines, which lines yield, therefore, points Fig. 12. for the spinodal curve. Here again three plaitpoints are to be expected: 1st. a realisable plaitpoint P, aap: 152 ded above the curve : = 0, 294, a hidden plaitpoint P, on the left of av aw dp a —0 and above — =0, and 3". the ordinary gas-liquid plait- det Lv ( 841 ) Zas a7 ‘ point P, on the left of Re but shifted to the side of the small av" volumes. Now it is to be expected that the value of p in the first mentioned plaitpoint is smaller than in the last mentioned. For 7%, strongly rising, the pressure strongly decreases when we pass along 4? 5 2 to the right and only if the curve ~ =O should strongly Iv? da? av 42 wp : HE project above iar 0 we should enter the region of high pressures. av The hidden plaitpoint has, of course, far lower pressure than the two others. The value of a for the first mentioned plaitpoint is larger than that for the hidden plaitpoint. The gas-liquid plaitpoint has the smallest value of «. Proceeding along the spinodal curve we get a course of p, as has been previously drawn by me. (See These Proc. [2 March 1905 p. 626). If 7 is made to increase, hey contracts. av 2 The top moves to the right, and reaches a position, in which — v is negative for the critical circumstances. But this means that the gas-liquid plaitpoint and the hidden plaitpoint have coincided already before. When they coincide we have again, as we observed p. 744 A 5 Pp dv dv d?v d?v d dv dy Aft thi 4 Er = is bd rc — ae a Tey Soe == Se € > al e dx yp da q dx? p da? q “ da? i da ; el us com ciding we have again a simple plait with a simple plaitpoint. But the plaitpoint lies far more to the left than would be the case if the 42 curve = =O did not exist any longer, and it also has a much Ak larger pressure. With further rise of 7’ nothing of importance is to 2 be expected. For neither the fact that ee: lies quite outside av dw iy ; ‘ é b Fie = 0, nor that ao = 0 vanishes, gives rise to new phenomena, because this takes entirely place in the unstable region. If we now draw either the value of the plaitpoint temperature or of the plait- point pressure as function of , and if we restrict ourselves to the realisable quantities, so excluding the hidden ones, this line separates into two detached parts. The right part begins at that value of 2, in which the plaitpoint P, possesses a pressure large enough to show itself on the binodal curve of the plait whose plaitpoint is P,, and passes then to c= 1. The left part begins at «= 0, and disappears before P, and P, coincide, namely, when P, lies on the binodal line, of which #, is then the plaitpoint. ( 842 ) That what we have called hidden plaitpoints, can never exhibit themselves, requires no explanation. That what in general we have called realisable plaitpoints, need not always show themselves, may indeed be assumed as known from the former thermodynamic con- siderations about the properties of the w-surface — but yet it calls for further elucidation now that we examine the properties of stability and of realisability by considering the relative position of the p- and the g-lines. We shall, however, only be able to give this elucidation, when by treating the rule to which I alluded in the beginning of this communication, we have also indicated the construction of the binodal curve. To get a clear insight into the critical phenomena for the case that for mixtures between two definite values of « the critical point 2 falls in the region in which — is negative, we must again distinguish at two cases; viz.: 1. the case that already at 7'—= 7}, the curve d* dy al partly projects above 5 — 0, in which case already at 7= Ty, at v the two plaijpoints P, and P, are found, and 2. the case that at ry a? ) . . . . yh T == Tj, the curve e=, lies quite enclosed within — — 0. In av av fig. 13 the second case is represented. Now if for values Ip of 7’ >7;,, the top of =e dv 2 zen I? =Ôdoes fall within —— =O da? there must have been contact of the two curves at a lower 7, and intersection at a higher T. As long as the two curves do not yet touch, the spinodal curve is little or not transformed, and no other plaitpoint is to be expected as yet than the ordinary gas-liquid plaitpoint which d, lies at smaller value of z than the top of fo If the two-eurves v intersect, the plaitpoints P, and P, have appeared first as coinciding heterogeneous plaitpoints, later as two separate ones. Naturally the ( 843 ) value of x for the two coinciding heterogeneous plaitpoints is larger than the value of « for the plaitpoint P,. With further rise of the aw : ; Typ temperature, when ass 0 rises further above PT 0, the plait- points P, and P, move further apart. P, moves towards larger values of «, and P, (the hidden plaitpoints) to smaller value of w. And the two heterogeneous plaitpoints ?, and P, coinciding at still higher value of 7’, there is a continuous series of values of z from «=O up to «=1, for which plaitpoints occur. For every value of only one. I have drawn (These Proc. VII, p. 626) the transformation of what I called there a principal plait and a branch plait. But this transformation refers, properly speaking, more to the binodal curve of such a complex plait than to the spinodal curve. If we then draw 7} as function of 2, such a line has a maximum and a minimum value, both lying above 7}. The minimum value at the origin of the double plaitpoint P?, and P,, and the maximum value at the disappearance of P, and P, in consequence of their coinciding. Also when /,; is drawn as function of x, we get a mil te dp dp dp\ dT dT, net ar curve. S neral — = — = s 7 simular Cu in genera is ee Tran re WI Je dt Av the value of /,; as function of Tj exhibits a more intricate form. dE dp (—) dele As ET _ : is determined by the proper- aT (4) do’ dT ERE da? Jr dy ties of the substance in the plaitpoint, e.g. by ae This quantity is Av pl cash (Rag i ’ d = 0, because (2) is equal to O in a plaitpoint. But T at the same for double plaitpoint, and so ze has two equal values in such a double plaitpoint. The plaitpoint curve has therefore two cusps in the case treated. The left branch extends from 7), to the temperature at which P, and P, coincide. The right branch begins at 7, and runs then back to the temperature at which the double plaitpoint P, and P, originates. The middle branch gives the hidden plaitpoints. But here, too, we must again notice that not the whole outside branches can actually be realised, the splitting up into three phases when we draw near the cusps having a greater stability then the homogeneous plaitpoint phase. These are the phenomena observed by. Kurnen for the mixtures of ethane and alcohols with greater values of 6 than that of ethane. Perhaps the change of ( 844 ) db Fi sas already account for the fact that the peculiar feature of at this phenomenon disappears more and more when, retaining ethane, we choose an alcohol with larger value of 4; so that the phenomena point to the fact that a normal plaitpoint line might be expected if 2 we proceed in the series of the aleohols. As condition for oa av being negative in the critical circumstances, we had: == 0 S . 837 a(1—-) 4b? 8 a = Gee Pee For in general it is to be expected that this value cannot so db easily be realised for large value of = than e.g. for almost equal ar value of 6, and 5,. That the mixture of ethane with methyl alcohol displayed quite different phenomena might already be expected on account of the fact that we have then a case for which with in- creasing value of 4 the value of 7). decreases. It is viz. almost sure that h is smaller for methyl alcohol than for ethane. dw If for 7’ = 77, the curve = —() should already partially project at a? ) . . . . above i —(), this will bring about but little change in the pheno- av menon. Only the minimum value of 7), will descend below T;, in the (7,1, 2)-figure. In the same way the left cusp will have to. be drawn at lower value of 7’ than 77, . It is, therefore, required for the course of the plaitpoint half mena, that 7, > Tj, and so according to the value of 74 (p. 830). Ma dx? (1 — 49) Sd, — «£(1— nk MEN een , lies between b In this inequality 2, dependent on the value of 5 3 1 1 1 Sige and y, between = and O. Let us write: a (1l1—a a, ( ) En)” a, 4- 4, — 201, ) (1—yy) 4 k Je ae ( 845 ) l 1 By successively increasing from a to —, and deriving the cor- b responding values y, and Er from the table of p. 829, we can 2 1 a, + a, — 2a,, a calculate the value which must have at the least in 1 Bie 1 ; b order to satisfy this inequality. If we put c= a to which ——— = 0 ait corresponds, we see that only a,=0O might be put. If z is made to increase, which implies that the ratio of the size of the molecules a, +a, — 2a, approaches 1, the value of * required to satisfy the in- a 1 1 equality, decreases. For the limiting case «= ive b‚=b, and y,=0, a, + a, — 2a 16 pl) + = must be > 57 10 enable us to put 7, > 7;,- a, But this value must be larger for 5, smaller than 6,, and the larger as the difference between b, and 4, increases. If this equality is not satisfied, so if 7, < 7%,, we have a plaitpoint line of a per- fectly normal shape. This is inter alia the case when for a low ratio between 6, and 6,, also a not very high ratio between the critical temperatures is found. First, however, we should have to know how a,, depends on a, and a,, before for given ratio of 6, and b, we a a could indicate how large the ratio of and a would have to be to 1 2 justify us in expecting either the complicated or the simple shape of the plaitpoint line. Moreover, 1 repeat that it should be considered in how far numerical values occurring in the given equations, would have to be replaced by others on account of the only approximate validity of the equation of state. From all this appears in how high a degree the properties of the 2 function influence the shape of the plait, and so also the miscibility & or non-miscibility in the liquid state, and that the influence of the properties of this function may be put on a level with that of the db ; JE function Ee We shall further demonstrate this by also examining av 2 : —0O exists, and intersects the curve the case that the curve dw ak dadv dx ( 846 ) el dy Let us now take a region of fig. 1 such that the line , == () av occurs on it, and that this line has the position as drawn in fig. 6. dw ; ; : Then the liquid branch of oe = 0 lies on the right side of the region Lv . dp r . at larger volumes than those of ees 0. These two curves might ie ; B , dy intersect on the left side. If now also the curve — = 0 occurs, v which will be the case if the temperature is low enough, and if ; ; dw Py this curve intersects both —— = 0 and ee 0, we have the shape atarv av of the g-lines as drawn in fig. 6, and there will again be formed a complex plait, whose shape and properties we shall have to examine. rn Ze=0 Strik-p-lijn = loop-p-line. Fig. 14. ( 847 ) J2 2 dp dp That intersection may take place of ze =—= 0 with dE teas an? dudv been proved on pages 834 and 835. We saw before that one gq-line may possess 2 or 4 points of contact with p-lines, but now we have a case in which the number of points of contact can rise to 6. In fig. 14 has been drawn: 1. the curve dp dp 8 = = 0 and = = 0, 2. the loop-p-line, 3. a g-line to which horizontal v av tangents may be drawn in 4 points, and a vertical tangent in 1 point and 4 portions of 6 p-lines touching the g-line. The pressure in point 1 is much larger than in 2, rises then, has a maximum in 3, descends again and reaches in 4 its lowest value. The greatest pressure is found in point 5, and in 6 the pressure has been drawn lower than in 5, but it may be higher than in point 1. Consult fig. 1 for the direction of the p-lines in the points of contact. These 6 points of contact are again points of the spinodal curve. So on the right there is again a portion of the spinodal curve which follows closely the zt OP. Wich by He line — =0 in its course, also on the left a portion that does not ae move far away from this line. But between these two portions the spinodal curve must have been strongly forced back towards smaller Gat volumes to avoid the line is ==). ak ; dp N =e In the points where tah intersects the curve = 0 the Av dy dp dadv spinodal curve touches this curve, because zr must be! = EN At C dv? for the points of the spinodal curve, and so it must remain in the 2 2 region where SEE is positive, except when =—= 0. It may ‘then wv y rar even be doubted if v >>b is found for all the points of the spinodal curve. Values of » <5 would mean that the left part and the right part of the liquid branch of the spinodal line would remain separated from each other; and this would imply for the miscibility or non- miscibility of the components that at the temperatures for which this is the case, even infinitely large pressure would be insufficient to bring about mixing. Already in my Théorie Moléculaire I raised this problem, and I showed, that if 4 is a linear function of .7, cases 58 Proceedings Royal Acad. Amsterdam. Vol. IX. ( 848 ) are conceivable in which the spinodal line could intersect the line d*b ; v= 5b twice, but that if oe has positive value, as is really to be aL expected, intersection will never take place. But if we acknowledge again that the knowledge of the equation of state is insufficient for very small volumes it follows that we had better not pronounce the solution of this question too decisively. If the spinodal line is closed on the side of the small volumes, then a realisable plaitpoint will be found there, while there must be a hidden plaitpoint in the neighbourhood of the points 2 x and 3. If the temperature is raised, the line ee U =), AS it nom 2 2 w = 0, can contract to above ae = 0; av also intersects the line adv 3 d before disappearing. If it has sufficiently ascended above — =0, v the spinodal line will get a point where it splits *) up, at which 2 new plaitpoints (homogeneous ones) are formed. So at this ‘splitting dy dv : : oe point — = 0 and —— = 0. This furnishes an indication as to the dx*P da", place where this splitting point will lie. That the g-line below dy dadv (p. 736), where we derived a series of points of inflection of the 2 d q-lines passing through the point in which ad =—0 has the greatest & = 0 must have a point of inflection, has been shown before volume. We have also previously (p. 628) derived a series of points d of inflection of the p-lines starting from the point where Ee 0 & dp j 8 and = = 0 intersect, and passing through the point where == 0 av 5 has the minimum volume. From this we conclude that the double d, Dah plaitpoint can only occur when the line = = 0 is intersected by id aw irae O pretty far to the left of the point with minimum volume, U d and so not far to the right of the asymptote of the line ik 1) This splitting point I had already in view in my Théorie Moléculaire (Cont. II p. 42 and 43) where I indicate the temperature at which the detached plait (longi- tudinal plait) leaves the v,v-diagram, when it has not contracted to a single point. ( 849 ) I may remark in passing that VAN per Lee’s observations for water and phenol illustrate the case discussed here, and that through the existence of a maximum pressure the properties of the vapour-liquid binodal line give evidence either of the occurrence of the asymptote of the line ze O in the v‚z-diagram, or of its lying not far to the left. So there are + plaitpoints after the appearance of this double plaitpoint. So two serve as plaitpoints of the plait which is detaching itself and they are both realisable according to our nomenclature and when detachment has taken place, both can actually be realised. They serve then as plaitpoints of what must properly be called a longitudinal plait. The two other plaitpoints, viz. the hidden plaitpoint which we placed in the neighbourhood of the points 2 and 3 above, and the lowest of the newly formed plaitpoints then form a couple of heterogeneous plaitpoints, which do not show themselves on the binodal curve of the vapour-liquid plait and will soon coincide and then disappear. From this moment the binodal lines of the two plaits are quite separated and behave independently of each other. The vapour-liquid plait, is then simple and perfectly normal. But also the longitudinal plait may then be considered as a normal one. (To be continued.) Waterstaat. — “Velocities of the current in an open Panama canal.” By Dr. C. Lexy. (Communicated in the meeting of March 30, 1907). $1. After an elaborate investigation the American Government has resolved on the execution of a project of a Panamacanal at high level, viz. at a height of 85 feet (25.9 M.) above the mean sea level. It will have three flights of locks. Against this project of the minority of the board of Consulting Engineers of 1905 there was a counterproject of the majority which favoured a canal at sea-level or rather a canal with one pair of locks. This canal would have been provided with one pair of locks in order to separate the Atlantic Ocean from the Pacific, but for the rest it would have been in open communication with these seas on both sides of the locks. | As a matter of fact this canal would not have been an open canal, therefore, like the Suez Canal, but a canal in which in most cases, 58* ( 850 ) if not in all, lockage would be necessary. A canal, therefore, which probably would have resembled more closely to the lockeanal pro- posed for Suez but not executed and strongly opposed, than to the present open Suez-Canal. The question therefore presents itself whether the Panama-Canal, like the Suez-Canal might not have been made open and without sluices. The technical commission of the International Congress of Paris in 1879 deemed a lock near the Panama-terminal an absolute neces- sity, because it was supposed that, without it the tidal motion of the Pacific would cause currents in the canal of a velocity of 2 to 2.50 M. per second *). On the other hand the Board of Consulting Engineers of 1905 rightly judged that the necessity of such a lock was not established but, owing to lack of time, it was not able to investigate the matter ’). On page 56 of the report we find as follows : “The question of the necessity of a tidal lock at the Panama end “of the canal has been raised by engineers of repute, but the limited “time available to the Board has not permitted the full consideration “of this question which is desirable. It is probable that in the “absence of a tidal lock the tidal currents during extreme spring “oscillations would reach five miles per hour. ‘(2.24 M. per second)” “While it might be possible to devise facilities which would permit “ships of large size to enter or leave the canal during the existence 1) This opinion clashed with that of the original projectors Messrs Wise and Recius. In a statement made by the latter at the meeting of the Technical Commission of May 19, 1879 he explains that the inclination of the high and low waterlines in the Panama-Canal will be about the same as on the Suez-Canal, as a consequence whereof velocities of the current might be expected in the Panama- Canal which would not exceed very appreciably those of the Suez-Canal. The latter, as far as they are due solely to the tides, usually do not exceed 0.90 M. per second; under the influence of wind they may increase to 1.30 or 1.35 M. 2) At the time of the meeting of the Consulting Board competent experts were sull of opinion that a lock at the Pacific-terminal would be necessary. Such appears clearly from the letter of Mr. T. P. Sports Chairman of the Isthmian Canal Commission received by the Board at the beginning of its labours. In this letter occur the following lines: “A disadvantage which the two plans have in common is that the rapid develop- “ments of naval architecture make it difficult to determine the proper dimensions “of the lock chambers. It is to be considered, however, that up to the present “time such developments has not been greatly hampered by deficient depth in the “harbors of the world, and that development here after will have that obstruction “to contend with. Moreover, it is not possible to dispense with locks entirely. Even “with the sea-level canal a tide lock will be required at the Panama end”. ( 851 ) “of such currents, the Board has considered it advisable to contem- “plate and estimate for twin tidal locks located near Sosa Hill ‘even though the period during which they would be needed would “probably be confined to a part of each spring tide.” It would require a special investigation, however, to know whether in a canal provided with locks, those locks would have to be used only during part of the spring tides. For, the oscillations of the sea above and below the mean level executed in a period, of three hours are on an average + 1.23 M. at neap tide and + 2.53 M. at spring time. This being so it seems probable enough that, both in the interest of navigation and to prevent eventual damages which might be caused by the closing of the lockgates against a strong current, lockage of the ships would be preferred to passing the lock with gates open. For, assuming the total profile of the locks to be equal to the profile of the canal, observations made in the Suez-Canal justify us in evaluating the velocity of the current at 0.70 to 0.90 M. at mean neap tide and at 1.00 to 1.30 M. at mean spring tide. At all events, each time after the gates having been closed the passing of the lock with gates open would not be possible before the sea had again reached its mean level. As a consequence, at each tide requiring the closing of the gates, the period during which passing of the lock with open gates would be possible, would be less than three of the six hours included between two returns of the sea to its mean level. Howsoever this be and leaving out of consideration the question to what degree a lock in a sea level canal will be an obstacle to navigation, it appears at all events that the necessity af such a lock has remained an unsolved question when in 1905 the projects of a Panamacanal were examined. The cause thereof lies in the uncer- tainty about the velocity of the currents which will occur in an open canal, particularly as a consequence of the tidal motion of the Pacific. In addition to the motion caused by the tides, great velocities of the current may occur in a sea level-canal, with or without tidal lock, at the time of high floods of the Chagres and other rivers, if the water of these rivers must be carried off by the canal. In contradistinction to the project of 1879 such would have been the case in the sea-level canal according to the project of the Board of Consulting Engineers. The Board comes to the conclusion that in a sea level canal with tidal lock currents will thus be caused reaching a maximum ( 852 ) velocity of 1.148 M. per second. (2.64 miles per hour). The Board is of opinion that such a velocity will be no hindrance to navigation. These same velocities will occur in an open canal as well as in a sea-level canal with tidal lock, at least if in both cases the water of the rivers must be carried off by the canal. They occur very rarely however and need not necessarily lead to an increase of the maximum velocities caused by the tidal motion. $ 2. The reasons which led the technical commission of the Congress of Paris in 1879 to expect currents with a velocity of 2—2,50 M. per second in an open canal, are twofold. In the first place, the commission gave some examples of currents with a velo- city of 2—3,50 M. per second observed on the lower course of rivers where similar differences exist between high and low-water as on the Panama canal on the Side of the Pacific.*) In the second place the commission published a memorandum of Mr. Kreirz, one of its members, containing some summary calcula- tions in regard to the velocities which must be expected in an open canal.’) It is evident that on the lower end of a river with a great amplitude of the tide very considerable velocities of the current may occur; but it does not follow that equal velocities will occur in an open Panama-canal. This will be the case only if the remaining circumstances which have a decisive influence on the velocity are about the same in the two cases. Now it is evident that the velocity of the current caused by the tidal motion of the water will be no less. dependent on the depth and in particular on the mean depth for the whole of the width, than on the amplitude of the tide and this irrespective of the question whether we have to do with a river or with an open canal of relatively great length. In other words the velocity of the current will depend as well on the proportion of the amplitude to the mean depth as on each of these quantities separately. Furthermore it is easily seen that in a river these velo- cities will depend in a great measure not only on the discharge but also on the changes of width and depth and on the inclination of the bottom near its mouth. In fact, the examples communi- cated by the commission show clearly that the velocity must be dependent in a high measure on other causes besides the amplitude of the tides. For among the examples of the commission we find the Rivière de VOdet with an amplitude of the tides of 5 M. and 1) See: Congrés international d’études du canal interocéanique. Compte rendu des séances. Paris 1879 page 362. 2) Ib. p. 384 and Pl. IV fig. 6. ( 853 ) veloeity of the high water flow of 3.50 M. and furthermore La Charente with an amplitude of the tides of 6.35 M. and a velocity of the high-water flow of 2 M. As regards the calculation of Mr. Kusirz, it is as follows: ATLANTIC PACIFIC. B. High-tide level. d= 385M* Mean sea-level. daer MNM {S93 24M... hs HAU Me Low-tide level. ad Ja195 Ms Se pn heek 73 660 M; none se | oe According to the above figure the area of the wet section of the canal on the side of the Pacifie was adopted to be 385 M.* at high- water and 195 M.* at low water. The difference between the mass of water in the canal at high and low tide is then taken for the volume of the prism ABC. 385 — 195 therefore = 73000 « EET M? = 6.935.000 M*. As the interval between high and low tide is about six hours, the change of the mass of water per second is found to be 321 M® on an average. The mean wet section of the canal on the side of the Pacific being '/, (385 +195) M’, that is 290 M’, Mr. Krerrz derives for the velocity of the inflow during the whole period of high tide or for the outflow during the whole period of low tide 321 eee lst 290 Furthermore, assuming that the most rapid change of the mass of the water will occur about at the time at which the sea-level is ( 854 ) equal to the mean level and besides, that this most rapid change is equal to double the mean change, the maximum inflow is put at 2 ><.321-= 642 MP. As the wet section of the canal at the mean level is about 277 M?, we find (aa = 2.32 M. per second for the maximum velocity. It is easily seen that these calculations are valueless. For the fact has been wholly overlooked that a certain time must elapse before some rise or fall at the mouth of the canal on the Pacific will make itself felt over the whole length of the canal. If therefore, shortly after ebb, the level in the canal near its mouth begins to rise and, shortly afterwards, the first inflow takes place, the level of the canal further inland will still be falling and the water will there be flowing out as a consequence. Similarly when shortly after the moment of high tide on the sea, the level of the canal near its mouth begins to fall and shortly afterwards outflow sets in, the level further inland will still be rising and there the inflow will not yet have ceased. Moreover the in- and outflow of the canal on the side of the Atlantic has been left wholly out of consideration. They will certainly not be small but will not take place at the same moments as the in- and outflow on the side of the Pacific. We may see that the difference in time, before mentioned, will not be insignificant but will have a great importance, by considering that, on the Suez-canal, the propagation of the high tide takes place with a velocity of about 10 M. p. second. Assuming the same velocity for the Panama-canal the propagation of the tidal motion over the whole of the length of the canal will require about 2 hours. As a consequence the currents near the two terminals of the canal will have different directions during a great part of the tide. The incorreetness of the reasons for the conclusion of the congress of 1879, according to which a lock is to be considered an absolute necessity seems to have attracted little attention at that time, and consequently the canal was originally executed with the intention of building a sluice on the side of the Pacific. FERDINAND DE LessEps, who always considered it a great advantage that the Suez-canal was executed without locks, probably never favoured this loek in tbe project of the Panama-canal. This led him in May 1886 to address himself to the French Academy of Sciences, requesting it to institute an investigation about the influence of the tidal motion of the Pacific and the Atlantic on the motion of the water in an open Panama-canal. The commission charged with this investigation reported on the ( 855 ) matter in the meeting of 81 May 1887. This commission consisted of the members of the section of Geography and Navigation and besides of the members Daupriz, Favé LALANNE, DE JONQUIERES and BoussiNksQq and the reporter BouqureT DE LA GRIJE, This report, though short, contains the results of extensive com- putations, which led the commission to the following highly remar- kable and important conclusion. “que, dans aucun cas, les courants dus a la dénwellation ne pour- “pont depasser 24 noeud” (+£1.29 M. par seconde, ‘et que cette vitesse, “qui ne peut être atteimte tous les ans que pendant quelques heures, “ne parait pas de nature a géner la navigation des bateaua à vapeur “dans le canal que Con creuse actuellement à Panama’. This conclusion was accepted by the Academy and the question concerning the possibility of an open Panama-canal without Jocks was placed in quite another light than that in which it appeared after the congress of 1879. Owing to particular circumstances, this conclusion of the French Academy of Sciences has attracted comparatively little attention. For in the same year that this conclusion was reached, the original project of a sea-level canal with loek had to be given up and to be replaced by a canal with several locks. It was the beginning of the sufferings of the Panama-canal. Since then the principal consideration has always been to limit the excavations to the utmost. For this purpose the hilly country required a canal at high level, consequently several locks. § 3. Therefore, if we wish to answer the question whether an open Panama-canal without sluices is possible, we have to inquire in the first place, whether the report of the French Academy of Sciences, of 1887 is based on sound foundations. What were these foundations ? In accordance with observations at the tide-gauge at Panama the differences between high and low water, in other words, the ampli- tudes of the tides at the mouth of the Panama-canal were adopted to amount to: at neap tide, on an average 2.46 M. 2? spring > 2) 9) 9) 5.06 M. a » 4, maximum in March or Sept. 6.76 M. The commission now calculates the velocity of the current for this maximum difference in height of the tides on the Pacific of 6.76 M., ( 856 ) neglecting the usually small tidal oscillation in the Atlantie and further starting from the following suppositions : 1. that experience shows that on a canal communicating on the one side with a sea of variable level, on the other side with a sea of constant level, the amplitude of the tidal curve diminishes uni- formly from one sea to the other and further that the retardation of the tide is proportional to the distance, that therefore : if Y = half the amplitude of the tides of the Pacific, /= length of the canal, w = velocity of propagation of the tides, the level y, with respect to the mean canal- or sea-level, at a distance x from the Pacific, will be: 8 x x y— — Y{ 1 —— | cos. | 2t — — ‘ l w 2. that, in accordance with what has been observed on similar canals, particularly on the Suez-canal between Suez and the Bitter- Lakes, the velocity of propagation of the tide can be represented by the well known formula : o=/ (tg) ke H = depth of the canal below mean sea-level, where : v = velocity of the current, K == constant (0.4 at flood-time, 1.2 at ebb); 3. that, from the levels which have been derived by means of the suppositions 1. and 2. for any moment and for two mutually not too distant places, the velocity of the current for that moment may be computed by applying the formula : » = 56,86 Ri — 0.07. By means of these suppositions the velocity of the currents have been computed for places at 9, 27, 45 and 63 K.M. from the Pacific, assuming a tide of the amplitude of 6.76 M. The results are as follows *): 1) The length of the canal which according to the project made at that time, would amount to 72 K.M. has been put at 76 K.M. in the calculations to allow for the curves. The bottomwidth was put at 21 M, the depth at 11.50 M. below mean sea-level at Panama, and 9 M. at Colon, the slopes at 1 horizontal on 1 vertical. Time alpend Distances from the Pacific. | since-low tide SSS on the Pacific.) 9 K.M. | 27 K.M. | 45 K.M. | 63 K.M. Moorthy s. | Velocities of the Sir ent in M. per second. 0 Weens 1200. 1. 0 hedde } EB 0590 |” — 0, 930-72 1 — 0.60 (abe 0.84 — 0.87 | — 0.83 1: ee 075" | — OBR Ba 2 + 0.35 | — 0.59 | — 0.75 | — 0.86 24 | + 0.67 — 0.34; — 0.63 | — 0.81 3 + 0.84 + 0.35 | — 0.42 | — 0.73 33 + 0.93 | S063) | 0,08) == 10:61 4 MOE Id 0.78 | = 0.487) =O: 4} + 1.02 | + 0.93 | + 0.80 0 5 le 4-06 | = O2ee | O58 53 et Gr) S144 | 20.86 | 0:60 6 DONE 1.06 | 40.98 | == 0.76 63 a 97 |" ot | “He 0:97. | HOES current from the Pacific lon the Atlantic + — “ 5 „Atlantie nr Aen From these computations follows that the maximum velocity in* the canal on the side of the Pacific, due exclusively to the tidal motion, will amount to 1.17 M. Supposing that there might be some difference between the mean sea level of the Atlantic and the Pacific and that this difference might amount to 0.50 M., the com- mission concludes that the maximum velocity might then increase to 1.26 M. The commission thus finally arrives at the conclusion referred to above. § 4. The two first suppositions on which the computations are based will probably not seriously deviate from the truth. For they are, at least partially, confirmed by what is observed on the Suez-canal. The commission further points out, that the formula for the velocity of propagation of the tidal wave, which has been derived in the supposition that the amplitude of the tide is relatively small as compared with the depth of the water, leads to results which, for the Suez-canal, agree closely with the observations. For the formula ( 858 ) leads to a velocity of propagation of 10.06 M., whereas we find 9.54 M. by observation. Matters stand somewhat differently for the third supposition. The formula by which the velocities of the currents are computed is the well known formula for permanent uniform motion. It is in the nature of the thing that such a motion cannot occur in a canal where a strong tidal motion takes place. But the question on which every thing depends is not so much this, whether the use of this formula leads to sufficiently correct velocities for any moment, as the following, whether the computed maximum velocities are not too small. In reference to this question we may remark that in general the formula will lead to too small a value of the velocity during the period that change in level is accompanied by decrease of inclination ; to too great a value where the change is accompanied by an increase of inclination. If, taking this into consideration, we examine the parts of the canal K.M. O—9 and K.M. 9—27, during the period of 4'/, to 6 hours after low tide on the Pacific, we get as follows: Time elapsed Mean inclination sincelowtide K.M. 0—9 | K.M. 9—27 4} hours 0.000044 0.000040 5 pe 0.000048 0.000046 har an 0.000048 0 .Q00047 alg | 0.000044 0.000045 | From these data it appears that, during the half hour preceding the epochs at which the velocities reach their maximum value at K.M. 9 and 27 the mean inclination for the part O—9 as well as for the part 9—27 has been little variable but increasing. From this it follows that by the application of the formula at these epochs we probably cannot have made any important error. Meanwhile, in order to test the validity of the computations, we have still to inquire whether the computed velocities, taken in con- junction with the computed levels, satisfy the equation of continuity. di dv dl we Tides de where J represents the area, v the mean velocity of the wet section at the distance z from the Pacific, at the epoch tf. ( $59 ) We can make out, approximately, in how far the computed levels and the velocities satisfy this condition by availing ourselves of the levels and velocities computed for each half hour and for the different distances from the Pacific. We thus find as follows: A. For the differences in the discharge at 9 and 27 K.M. distance from the Pacific. Moon-hours Area Velocity: ice Per half hour in | Excess of elapsed since I Vv Iv part 9—27 inflow over low tide on the, = 5 = | outflow in Pacific. SH abe Ad a Te Pake in flow out flow half an hour M?.| M2.) M. | M. | M3.) M3. M°. M'. M'. At 450} 388/1.02,0.93| 459) 361 914000 715000 201000 3 475| 40711 .1711.06| 556} 43 1.013000 807000 --206000 Di 491} 4201.161.14} 570) 466 1 .002000 828000 +174000 6 498) 428 1.09 1.06 543) 454 | | B. For the change of the mass of water contained in part 9—27. Moon-hours Area Change of area Mean Changeofmass } during half | ; elapsed since I an hour. change | per half hour low tide on the — —| for part for part Pacific. 9 | 27 | 9 yale) ld ek vd ae i M) M2] M2 | M2 M’. M*. Ak 450 | 388 | to pls ar ji HDD + 396000 5 415 | 407 | +16 | +13| + 445 | +'96f000 54 491 | 420 eet 8} Soe le a 6 498 428 Comparing the last columns of following differences for part 9—27 : from 4'/, to 5 hours + 195000 M*, or on an average per sec. + 108 M?’. 5 Tie AS AO Ke dele) 9) 9 „9, -— 9) 9) 55000 „ 39000 ,, 2 9) kh) 99 399 39 9 > the tables A and B we get the Pes raphe ere 22 99 be) 2, bP) It appears from this comparison, that by the computed velocities, taken in conjunction with the computed levels, the condition of continuity is not fully satisfied. Therefore, assuming the levels to be correct, the velocities need some correction, ( 860 ) Suppose these corrections for the consecutive half hours to be for K.M D= dd dend for KM. Mdd Onde we find for the values of the corrections: d, = + 0.15 M. d= — 0,13 M. d, = + 0.12 ,, B = —0,10 „ d, = — 0.04 ,, de = +004 „ d,= — 0,01 „ d/=+0,01 „ Therefore, applying the corrections, for the velocities themselves : at K.M. 9 at K.M. 27 at 4°/, hours 4 7M. 0,80 M. oo ' A 0,96 ,„ te ee Bae A die Bs ee ae - 108. 3, 1 A ap From these numbers it appears that we can satisfy the condition of continuity at least for the part 9—27, during the period between 4'’, and 6 hours after low tide, by relatively speaking slight modi- fications of the computed velocities. It cannot be denied, however, that the circumstance of the condition of continuity not being necessarily satisfied in applying this method of computing the velocities, indicates that this method is uncertain to some extent; though it appears that the uncertainty, at least as regards the calculation of the maximum velocities, will be small. Another reason of uncertainty in the computation of the velocities lies in the value assumed for the coefficient of the formula for uniform motion. This value, 56,86, is not the result of a great number of obser- vations made on rivers and canals of about the same inclination and depth as the Panama-canal, but of observations for rivers of considerably smaller depth. We may of course test the validity of this coefficient, as well as, more generally, the validity of the formula itself, by comparing the velocities it yields for the Suez-canal with those really observed there. Of the observations which have been made about the velocities in that part of the canal which lies between the Bitter Lakes and Port-Thewtik, those of 23 July, and 8 en 22 August and 6 September 1892 have been published *). These observations, however, are insufficient for a fair comparison. 1) See: The Suez-canal according to the posthumous papers of I. F. W. Conrap arranged by R. A. van Sanpick. Tijdschrift Kon. Instituut van Ingenieurs 1902—1903, p. 89 and 90. 4 ( 861 ) They have been made for two parts of the canal each 200 M. in length and separated by only 4.9 K.M. One part was included in that division of the canal which at that time had been widened to a bottomwidth of 37 M. while the other, having a bottomwidth o only 22 M., was situated a little beyond the point of transition to the not yet widened canal. As a consequence the motion of the water on the whole of this part of the canal, 4.9 K.M. in length, cannot have been uniform *). Moreover these observations are only relative to the velocities in the middle of the current, observed by means of floats down to a depth of 6 M. below the surface, whereas the velocity given by the formula represents the average velocity for the whole of the wet section. Meanwhile a comparison of these observations with the results obtained by the formula might still give some idea about the reliability of the formula. The comparison of the observations referred to above with the results yielded by the formula, putting the coefficient at 56.86, lead to the following results: > OBSERVATIONS ON THE SUEZ-CANAL IN 1892. Direc- Distance pee Observed velocities. | Computed between level He. Averages during an | mean velo- Day and hour of | tion of |the Pla tween the hour in the sep ortie ces 0 i the observation the cur-\Opserva- glas of widened rent. tion. obser- | widened | unwide- E vation. part | ned part} Part.*) K.M. M. M. M. M. 23 July 11—12a.m. | flood 4.9 +012} +0.75 | + 0.97 | + 0 64 os 5—6 p.m. | ebb 4.9 — 0.14} — 0.84; — 1.11 | — 0.58 8 Aug. 11—12a.m. | flood 4.9 + 0.09 | + 0.69 | + 0.87 | + 0,47 a 5—6 p.m.| ebb 4.9 — 0.41 | — 0.80 | — 0.93 | — 0.57 i ale 6—7 a.m. ebb 4.9 — 0.16 | — 0.88 | — 1.05 | — 0.68 » y» 12-4 pm.| flood | 4.9 | + 0.07} + 0.66) + 0.82 + 0.46 6 Sept. 11—12a.m. | flood 4.9 + 0.07 | +066) +089) + 0.47 sai 5—6 pm.| ebb | 4.9 — 0.10} — 0.85 | — 0.98 | — 0.53 1) The first part was the widened part of the canal between K.M. 149 and 149.2; the other the not widened part between K.M. 144,1 and 144.3. The tran- sition of the widened to the not widened part was situated at K.M. 144.4. 2) As the part of the canal from K.M. 149 to 144.4 had been widened the observed difference of level is relative to the widened part. ( 862 ) From this table we derive for the proportion between the computed average velocity for the whole of the wet section to the velocities observed down to 6 M. in the middle of the widened part of the canal, the following values: at high water flow at ebb flow (from the Red Sea) (towards the Red Sea) a i Af 1.45 1.47 1.40 1.43 1.30 1.40 1.61 Mean 1.37 1.44 The true value of this proportion for the case in which observation and computation agree, is unknown. But if we consider that the floats went down to only 6M. below the surface, whereas the depth of the water at flood tide was over 8.50 M. and at ebb time over 7.50 M. and furthermore, that the canal had side slopes of 1 vertical on 2°/, horizontal, we conclude that at all events the velocity in the middle must have considerably exceeded the average velocity for the whole of the section. As far as can be ascertained therefore, the formula applied to the Suez-Canal leads to results which do not clash with the observation. More conclusive information cannot be derived from a comparison of the computed velocities to the observed values. As long therefore as complete observations, made for the widened Suez canal, concerning the relation between the velocity of the current, the tidal motion and the dimentions of the section, have not furnished us with more reliable information about the value of the coefficient and about the question whether the formula applies fully to the case, we cannot avoid a relatively considerable uncertainty in the calculation of the maximum velocity. § 5. A closer examination is therefore required to decide in how far the velocity of the current in an open canal may cause a hin- drance to navigation and whether this hindrance cannot be overcome. In discussing this question we must consider, on the one hand that the computed velocities represent average velocities for the whole of the wet section and that therefore the absolute velocities in the middle of the canal will be more considerable; on the other hand, however, that the computed velocities are relative to the greatest possible differences in the height of the tide. The computed maximum ( 863 ) velocities may occur therefore only on a couple of days every year. And on these days only during a few hours. In how far a relatively rare velocity of the current offers diffi- culties to navigation is of course ascertained in the best way by a comparison to canals on which under similar conditions similar velocities occur. For such a comparison the Suez-canal offers the best conditions. For this canal several observations about the velocity of the current are known. Published observations, however, cannot lay claim to completeness, at least not for the present purpose. In the first place because they have not been frequent enough to justify the belief that among them will have occurred these rare cases which by an unfavourable coincidence of circumstances, must have given rise to exceptionally great velocities. In the second place because the measurements are, as a rule, relative to absolute velocities in the middle of the canal and not to the average velocity for the whole of the wet section. Moreover, in comparing the Panama-canal to the Suez-canal we have to consider that the dimensions of the former will be much more considerable than those of the latter as originally executed. Consequently such velocities as have caused no difficulties for the Suez-canal will cause them still less for the Panama-canal. For the Suez-canal between the Bitter Lakes and Suez originally had a bottomwidth of 22 M. and a depth of 8 M. below mean springtide low water, with which dimensions corresponds a cross section of 3830 M?*. On the other hand the sea level Panama-canal would get a bottomwidth of about 45.7 M. (150 feet) and a depth of about 12.2 M. (40 feet) corresponding with a cross section of 855 M?’. Observations, made during the period 1871—1876, have brought to light the following facts about the velocities of the current in the Suez-canal between the Bitter Lakes and Suez. *) “The maximum velocity of the high water flow, running North- “ward, amounts to 0.80 to 0.90 M. at the springtides of the months “of May and November, to 1.15—1.35 M. p. s. in the months of “January and February. “The maximum velocity of the ebb flow running Southward amounts “to 0.75—0.80 M. at the springtides of the months of May and Novem- “ber, to 1.20—1.25 M. p. s. in the months of July and August. “Along Port-Thewfik in the canal south of the main channel 1) Vide the paper of Mr. J. F. W. Conrap pp. 89 and 90. og Proceedings Royal Acad. Amsterdam. Vol. IX, ( 864 ) “towards Suez, bottomwidth 80 M., the velocity of the high water “flow at springtide is 0.60 to 0.70 M., at neaptide 0.45 to 0.50 M., ‘in the winter-season with strong South wind 1.00 to 1.20 M. p.s. “The velocity of the ebb flow at springtide is 0.55 to 0.60 M. In “the summer with strong North wind 0.90 M. p. s. “Outside the mouth of the canal at Port-Thewfik no velocity of “the current has been observed.” The observations of 23 July 1892 made under circumstances which, as regards the flow, were certainly not unfavourable, led already to velocities which, at flood tide, ranged from 0.95—1.03 M. and were in the mean 0.97 M. at flood tide and 1.11 M. at ebb. Mr. Davzars, chief engineer of the Suez-canal, speaking at the meetings of the Technical subeommission of the International Con- gress for the Panama canal in 1879, stated in regard to the sidings of the Suez-canal, as follows ©): “Dans les canaux où le courant est faible, et la où n’existe aucun courant, il suffit de faire les gares d'un seul côté; mais des que la vitesse atteint 0.75 ou 1.50 M, il faut les établir des deux côtés et en face Pune de l'autre”. By this statement we are certainly justified in concluding that the said engineer, founding his opinion on his experience of the Suez- canal, deemed allowable velocities of the current of 1.50 M. The small original bottomwidth of the Suez-canal of 22 M., however, caused difficulties for the simultaneous navigation in both directions. The following communications of Mr. E. QuELLENNEC, consulting engineer of the Suez-canal company, proves that these velocities of the current offer no difficulties even for the big ships which at present navigate the Suez-canal. These communications to the Board of Con- sulting Engineers of 1905 are as follows : “In the Suez section the velocity of the current very often exceeds 0.60 meter per second, and reaches at times 1.35 Meters per second. “In the latter case the ships do not steer very well with the “current running in; however the navigation is never interrupted “on account of the current. In the Port Said section ships can “moor with a current running in either direction; in the Suez “section they always moor with the current running out”. ’) The canal between the Bitter Lakes and the Red Sea has at present a width of about 37 M., but a widening of the cross section h See: Congrès international ete 1879, p. 361. *) See: Report of the Board of Consulting Engineers for the Panama-canal, Was- hington 1906, p. 176. ( 865 ) to 45 M. width and 10.5 M. depth is being executed. After this widening, navigation will certainly experience still less difficulty than at present. Meanwhile, and this point deserves attention, the velocity of the current after the completion of the widening for the whole of the canal between Suez and the bitter Lakes, will not be lessened but increased. For, owing to the surface of the two Bitter Lakes, which is about 23800 H.A., the widening will only cause insignificant modifications in the level of these Lakes. Consequently the fall of the water between the Red Sea and the Bitter Lakes will be nearly unaltered after the widening both at high — and low water. Under these circumstances the enlargement of the cross section will necessarily cause increased velocity of the current. The mere consideration of the maximum velocity which may occur during a few hours every year, and even then only on the side of the Pacific, is evidently inadequate for reaching a true estimate about the question whether the velocities of the current in an open Panama-canal without lock will offer difficulties of any impor- tance for navigation. We have to pay regard in the first place to the velocities which will regularly occur on the whole length of the canal at mean spring-tide and mean neaptide. These velocities may be derived with some approximation from those found by the French Academy for a maximum difference in tide of 6.76 M.”), at least if we suppose that these velocities will not considerably deviate from the truth. We thus find for the maximum velocities at K.M. 9 27 45 63 at mean neap tide: 0.70 M.. 0.67 M. 0.59 M.. 0.54 M. ee SPE ne AO <2 000740 4 OBD 5 0.74 ,, The following diagrams show the velocities of the current, for the interval of from 9 to 63 K.M. distance to the Pacific, at mean spring tide and mean neap tide, 0 to 6 Moon-hours after ebb on the Pacific. They were derived from the calculations of the French Academy of Sciences. 1) The approximation neglects the differences of the velocities of propagation of the tide for different amplitudes. We thus obtain for the velocity v’, at an arbitrary place, the amplitude being 4’, the following value, which is expressed in terms of the velocity v for an amplitude y: (v' + 0.07) = (v + 0.07) [yt | y ( 866 ) SPRING TIDE. NEAP TIDE. K.MM.65 45 ae 5 Velocity. G3 45 ae OKE 1oom 5 3 2 = 3 +0.50M = Ss 3 = 2 — 2 +0.25M 5 = sa le) —o.25M 8 2 a —— +-0.50f = Q ia a +o 7s nr = too M The figures inscribed in the diagram represent the hours elapsed since low tide in the Pacific. + —= current from the Pacific towards the Atlantic a te A „… Atlantie » acini: § 6. From the preceding considerations we may conclude that, as far as we can judge by direct computation of the velocities, to be expected in an open Panama canal, there is reason to think that these velocities will indeed be somewhat, but not considerably greater than those on the Suez-canal between the Bitter Lakes and the Red Sea. Meanwhile we ought not to forget, that both in these computations and in our knowledge about the velocities which occur on the Suez- canal there remains some or rather considerable uncertainty. This uncertainty might only be diminished by more complete observations than have been published as yet concerning the relation between velocity of the current, tidal motion and dimensions of the cross- section of. the Suez-canal. We shall be enabled to get at a just estimate therefore about the question whether an open Panama-canal without lock is possible, only by following a way different from that of a comparison of the computed velocities with those observed on the Suez-canal. This way may consist in trying to get at a direct knowledge of the differences of the velocities on the two canals by a comparison of the circum- ( 867 ) siances which will oeeur on the two. Afterwards the circumstance that, on the Suez-canal the velocity of the current offers no difficulty, _in conjunction with the probable value of the velocity of this canal, will help us in deciding whether these differences are of such a nature as to produce undoubted difficulties on the Panama-canal. In making this comparison it will be permissible to assume that the violent winds occurring in the Suez-canal, which cause velocities of the current 0.30 to 0.50 M. in excess of those due to the tidal motions, are not to be expected on the Panama-canal near the Pacific. First, however, we have to inquire whether an open canal cannot be executed in such a way that for that part where the current will be greatest the difficulties caused by such great velocities can be removed. It is evident that this would be possible only by giving a very great width to the canal. This is practically impossible for that part of the canal which intersects mountainous country, but it is well feasible for that part of the canal which extends from the Pacific to the Culebra mountain, that is to near Pedro-Miguel, a part which for the greater part intersects low country. If to this part of the canal, where just the greatest velocities will occur, a bottomwidth is given of for instance 500 feet (about 150 M.) instead of 150 feet (45.7 M.) no difficulties will be experienced from any presumable velocity of the current. Such a widening of the canal on the side of the Pacific would however increase the inclination and the velocity of the current in the remaining part, at least if no particular measures are taken to prevent such increase. These measures would necessarily consist in making a reservoir or lake in open communication with the widened part of the canal. This reservoir or lake would have to be of such an area that it would be capable of retaining the water which, during the rise of the level, it would receive from the widened part in excess of what would be discharged by the unwidened part. During the fall of the level it would restore this surplus to the widened part. From the nature of the thing this arrangement is theoretically possible. Whether it be practically possible depends on the surface which a determinate widening would entail. A lake of somewhat over 800 H.A., such as is represented on Plate I, is feasible in the low country bordering on the canal near its mouth on the Pacific. Starting from this area it is possible to determine the degree of widening which may be given to the part near the Pacifie in such a way that, under given circumstances, for instance at spring tide, ( 868 ) no change will occur in the gradient of the high and low water lines, nor in the velocity of the current in the remaining part of the canal. As soon as the amplitude of the tides exceeds that of springtide the inclination and the velocity of the current will be somewhat in- creased for the wider part, somewhat diminished for the remaining part, as compared with what they would be without the widening of the first part and without the addition of a lake. In the case of a smaller amplitude of the tides the reverse will occur. Owing to the situation of the ground the junction of the widened canal with the lake must be made at a distance of about 12 K.M. from the Pacific terminal of the canal. Not before 3 K.M. farther however, that is not before 15 K.M. from the sea, the surface of the lake reaches a considerable breadth. Therefore if the inclination of the high and low water lines remains nearly unchanged and if, according to the most recent project, the length of the canal is fixed at about 80 K.M., the amplitude of the tide in the lake may reach (5.06 — 15 x 0.0632) M. = + 4.10 M. With such an amplitude a mass of water may be received, in the interval between high and low water, of 800 10.000 > 4.10 M*. = 32,800,000 M*. Assuming, as an approximation, that this mass is received within a period of six hours, we find that on an average 1500 M?. will be received per second. The surplus width of the part of the canal near the Pacific must be determined in such a way, therefore, that on an average 1500 M*. may be displaced — without increase of the velocity of the current — in excess of what might be displaced if the width remained normal. It is not well possible, without elaborate computation, to fix accu- rately the surplus width necessary for the purpose. But it is easily seen that this surplus width must be about 100 M. so that a bottom- width of 150 M. might be given to the widened part extending from the entrance of the canal to the junction with the lake. Corre- sponding therewith the width at the spring tide level would be about 250 M. At K.M. 64 this width might gradually be reduced to the normal width. It will be possible therefore to remove eventual difficulties offered by considerable velocity of the current on the part of the canal nearest the Pacific, by increasing the bottomwidth of this part. (16 K.M. in length). Now let us consider how the case stands for the remaining part of the canal, 64K. M. in length. ( 869 ) On this part the inclination of the high and low water lines will amount to 3.16 cM. at mean springtide and to 1.52 eM. at mean neaptide. On the Suez-canal the inclination of the high and low water lines between the Bitter Lakes and the Red Sea amounts to 2.52 cM. per. K. M. at mean spring tide and to 1.48 cM. at mean neaptide. Under the influence of the direction and force of the wind, the height of the tides on the Suez-canal may be increased or diminished by about 0.25—0,33 M. As a consequence the inclination of the high and low water lines may be increased by about 1 cM. per K. M. As the distance of the Bitter Lakes to the Red Sea is about 28 K.M., this already enables us to conclude that the velocities of the autem in an open Panama-canal, for the first 28 K.M. on the side of the Atlantic, cannot greatly differ from those which occur on the Suez- canal (See Plate II). . If therefore — leaving out of consideration the absolute value of the velocities — we may assume that the velocity of the current will offer no difficulties on the Suez-canal even when it will have been widened, then it follows that on an open Panama-canal, for about the distance of 28 K.M. from the Atlantic, no difficulties will be met with on account of the velocity of the current. Finally as to the middle part of the canal extending for about 36 K.M. between K.M. 28 and K.M. 64 from the Atlantic. For this part the differences between the velocities of the current, occurring therein, with those occurring in the preceding 28 K. M., may be computed with sufficient accuracy by means of the equation of continuity. For, let ab be the canal’s surface for this part, at the epoch t, a little before low water, at the distance of 64 K.M. from the Atlantic. Similarly let a’b’ be the canal’s surface a second later, then necessarily Aiken Ay Ay, 2 x 36000 x = ly — Iv, from which : 36000 (B +4 B,) (Ay + Ay) 4 Ll en v, —v) = ke (v, — 2) En - Now the quantities /,/,, B, B,, Ay and Ay,, are known for the epoch tf, at least if we admit that — as is the case on the Suez- canal — the high- and lowwaterlines for the part 28—64 K.M. are ( 870 ) Mean sea-level. 28. 64. ee Geese. Sia ee eee A \ ad Y, ' ' ' 4 Ee Area = 1. 4y Width at surface — B. Gr $ Velocity = v. — | | ' as nearly straight lines, and further that the velocity of propagation of the tides is known with sufficient accuracy, likewise owing to obser- vations made on the Suez-canal. Therefore we will be able to determine the difference of the velocities at 64 and 28 K.M. distance from the Atlantic, for the epochs at which Ee is a small quantity. 1 This will be the case near the moment of low water. Kor the difference of the velocities v, and v, during the half hour preceding the moment of lowwater at K.M. 64, during which half hour the velocity of the current will be maximum at that point, we find as follows for spring-tide. We assume that between the distances 28 K.M. and 64 K.M. (from the Atlantic) there is a retardation of the tides of just one hour : at '/, hour before lowwater: (v,—v) = 0.32 M. + 0.02 v v) = 0.12 M. + 0.015 v. From these figures it appears, that during the half hour before lowwater at K.M. 64 the differences of the velocities of the current are only to a small extent dependent on the value of the velocity v. These differences, therefore, may be determined with sufficient pre- cision, even if the velocity v is only approximatively known. By observations made on the Suez-canal during the period 1871— ,, lowwater : (v, ( 871 ) i me . * : 1876 the velocity-curve for a place near the Red Sea is known both for springtide and for neaptide. It has been represented in the following figures. *) Mean velocity-curves at the entrance of the Suez-canal. Springtide. Velocity. Neaptide. 1oomM SaaS e Ree eae EEE he 5 NP seek Ie oden tt MELA SR ieee ARS NEL z qa Ga Sah Ree eee MENE A EE NE EDE EEN bm 2, 5 EE AE EE les al gio 4 ab Se rae ® hey En wae BRaRRe AA EN AMEN NES MANE en DE SEE HEEN Ed AM RETE LOES 6.00: M x TARR SER al Ee BEERS ee eee, an RG? Rae eel ie || ES) (BMRSEB Beer, me ar Oe see Ree oe m > =—— OF SER RSS wie eer. te je ESE EEE | Bekele EN ogom a 8 LEISEL EREE sel El REN NEER oM ok 4 CG 2 | . . . \2. Ge ES Moon-hours after lowwater. a The above velocity curves probably do not represent the mean velocities but the velocities in the middle of the canal. They have been derived from measurements made every hour partly by means of floats partly by means of the current meter of WonTMann. It deserves attention, however, that at the time of these observations the Suez-canal had still only a depth of 8 M. below low water and a bottomwidth of 22 M. The section of the canal is now being increased to a bottomwidth of 45 M. and a depth of 10,5 M. below low water. The velocities in the widened canal may perhaps exceed by 20 percent those observed on the canal during the period 1871—1876.*) !) These curves are borrowed from the Etude du régime de la Marée dans le canal du Suez par M. Bourperres, in the Annales des Ponts et Chaussées of 1898, They occur originally in a Note sur le régime des eaux dans le canal maritime de Suez et & ses embouchures in 1884 by Lemasson Chief Engineer of the canal- works. 2) For the original cross section of 8M. depth below low water, 22 M. bottom- width and slopes of 1 vertical on 2 horizontal we have: Area [= 304 M?; wet circumference O=57.9 M., consequently R = 5.25 M. For the future cross section of 10.5 M. depth, 45 M. bottomwidth and slopes of 1 vertical on 21/, horizontal, we will have: J=749 M?., 0 =101.5 M. tberefore 7.37 R=1.37 M. Now GA = 1.20, 5.25 ( 872 )° If, in consideration of this fact, we substitute in the second member of the formula, for v the values observed in the period 1871—1876 increased by 20 percent, we finally find 1/, hour before low water (v, — v) = 0.33 M. at zon to SOME The differences 0.33 and 0.13 M. represent the differences of the simultaneous velocities, not those of the maximum velocities at the distances of 64 and 28 K.M. from the Atlantic. At the moment that the velocity reaches its maximum at K.M. 64, the velocity at K.M. 28, where the tides set in about an hour later, will still be below the maximum at that place. According to the observations on the Suez-canal we may assume that, at the epochs mentioned, the velocities of the current at K.M. 28 will at least be about 0.15 M. and 0.05 M. below the maximum of that place. Hence we may conclude that the maximum velocities at K.M. 64 and 28 will certainly not differ 0.18 M, and probably not much over 0.08 M. We are sufficiently justified therefore in assuming that the velocity at K.M. 64 may be about 0.15 M. in excess of that of K.M. 28. As appears from what has been stated before the difference is inferior to the increase of the velocity of the current on the Suez- canal under the influence of the wind, which may amount to 0.30—0.50 M. It cannot, therefore, cause any serious difficulty. $ 7. For an open Panama-canal executed as follows: From the Atlantic to K.M. 64 having the same normal cross section as that of the project for the sea-level canal ; from K.M. 64 to K.M. 68, which is the place where the canal will be connected with a lake gradually widening ; from K.M. 68 to the Pacific at K.M. 80 having bottomwidth of 400 to 500 feet; the following conclusions in regard to the velocities of the current at springtide may be accepted : On the first 28 K. M. of such an open canal, velocities of the current will occur at springtide which, on an average, will be about equal to those, which will take place at spring tide and with a moderate wind on the Suez-canal between the Bitter Lakes and the Red Sea as soon as the widening of this canal will be complete. On the subsequent 36 K.M. of such an open canal the maximum velocities at springtide will exceed those on the preceding part by SU | ) i | If ui | | | RN, 7 | fA Nth ACIFIC- HELL tp pe = x aN In PROFILE IE 1/1000. TALE 1/100.000. | } END OF LAKE. sHOOG:- WATER: SPRINGTI- 2.53 + | | (HÛAÂ | |= 12.19 M. BELOW LOW WATER ' 1 En eee, RS ABOVE MEAN SEA-LEVEL | Wins “LAAG WATER: SPRINGTY 2. ENE hi tr 2 5s ‘== HIGH WATER SPRINGTIDE. — LOW WATER SPRINGTIDE. BOTTOM OF LAKE. (Meerbodem). \ t | 74. ZG. 78. gok mM. C. LELY: “Volocitios of the current in an open Panama-canal’’. INI AL hey Fi . “PIHIVA> \ “PALO GRA SCALE 1,100,000, () SSE ER RO TAVERNIULA LAPLEDRA:SIN-CABEZA IN ) N NS DE y SS ASS Se CERRO:MAGUINCAL a 2 See SEN en or MÄCERRO-AGUA:ERIA: = EN Ce $ TECERRO Rico: AS | \ 4 2 CERRO:DE-LAS: MINA : BAAS RIO-GRAND | SSS 2 -RIO:GRANGE: . : | yo = 4 =e + MRA FLORES | eg CONE 1 NEN J NAT ro EERE é OBISPON7aE P 5 , 4 : 3 oe — *.MATACHIN- ( ns ef SET 2 ~ RS » y ‘GORGONA: Ge \ COLEBRA: a we Sr A, HRS é of Lt nAce\ f TU Oy NUR XC, é ° a + fs 5 =e ae IN = es Ying a Rio PERicoy, ANH Ae me wll EMPERADOR- CULEBRA: LONGITUDINAL PROFILE =) wb 5 VERTICAL SCALE 1/1000. = : HORIZONTAL SCALE 1,100,000, WY g fi ra f=} we hat a ; VA Fi Ei 1 Vp H -HOOG- WATER SPRINGTI 2. 53+ = HIGH WATER SPRINGTIDE. -OBISPO: \ PARAISO- MATACHIN: MIRAFLORES. END OF LAKE END OF LAKE Br ns UY Dj WM OO OFLA ELIS LEL DE ACE 5 | Zi ee | ze BOTTOM OF GANAL 40 FEET = 12.19 M. BELOW LOW WATER 5804) ‘ 1 | \ de B TT Sin ee SE 7 = - - } Ef 3 8 8 3, 8 3 EN El Ei 3 1 HEIGHT iN METERS ABOVE MEAN SEA-LEVEL | SE NE ES EG TG eS ; ; f | y 46 48 50 52 54 56 58 G8 zo #2 74 #6 28. 80.k rn Proceedings Royal Acad. Amsterdam. Vol, IX | HOOGWATERLIJN HIGH WATER LINE. So ees) | Aiea ONIN + 2.00 _ PANAMA CANAL + 1.50 zZ + OO 5 il z ati + 0.50 A LANTIC | | 5 PACIFIC —— all | | ES wu) p Oo = | U | = IAN LJ = D 5 8 « 050 © SI 3 ej EN dn | Z x ‚E q + 1.50 5 | < Na. | 2 + 2.00 oS | arke, re) 28 C4 goK.mM. LAAGWATERLIJN — LOW WATER LINE, CANAL DUNTRY Je854 M?2, entend or 2° Jot 230011? ie C. LFLY: “Velocities of the current in an open Panama Canal”. HIGH AND LOW WATER LINES AT HOOGWATERLIJN — HIGH WATER LINE = Z WIL KANAALVAK MET GROOTE BREEDTE a x EI SPRING TIDE SUEZ-CANAL ~~ (1871—1876) ee +100 = + too M en Alli. oso ral RED ATLANTIC +919 HOOG 4013 > ll MEDITERRANEAN —o En i EENES — : : So ats 0,01 Tost - 0.15 LAAG 0.12 { ae : II SEA OCEAN \ ' { \BITTERE:MEREN) |.) en 1 En. 5 pn aaa man B ry es ' | 1 ' H ' ts ! ' \ } HI ' | Ei THE HATCHED AREAS ARE +100 H L 1 1 sel tl RELATIVE TO THOSE PARTS + Loe ° Te 34 Gu 78 98 ; Bei wo co KM. OF THE TWO CANALS IN = BITTER LAKES 2 WHICH THE VELOCITIES OF 6 x ; E THE CURRENT WILL BE LITTLE +150 0 = i z DIFFERENT. < =I 2) r 4 x 8 Ie +200 b 5 lol : ew a Y 2 J +2.50 o 28 64 g0K.M P R O F | L E S LAAGWATERLIJN — LOW WATER LINE SUEZ-CANAL PANAMA-CANAL A. AS ORIGINALLY EXECUTED. A. IN LOW COUNTRY LOW WATER _— LOW WATER —-..—-- Je 854 MP LOW WATER -—_> (¢) LOW WATER —-.—.. SCALE OF THE PROFILES } 1000, Proceedings Royal Acad. Amsterdam. Vol, IX. PACIFIC ( 873) about 0.15 M. They will not exceed however those on the Suez- canal with a strong wind. For the last 16 K. M. of such an open canal the maximum velocities at springtide may be somewhat more considerable. On account however of the great width, which may be given to this part they will cause no serious difficulty. Therefore, if we assume, as we have good reason to do, that even at spring tide and with wind the velocities of the current on the Suez-canal offer no serious difficulty to navigation we may conclude that on a Panama-canal of the above description also navigation will experience no difficulties on account of the velocities of the current. Therefore, if we leave out of consideration the question whether an open Panama-canal without tidal lock is to be preferred either to a sea-level canal with such a lock, as proposed by the Board of Consulting Engineers, or to a summit level canal with three locks, as is now in course of execution, we may conclude, in the main in conformity with the conclusion of the French Academy of Sciences of 1887, but for different reasons: That the velocities of the current due to tidal motion in an open Panama-canal without tidal lock will be no obstiuction to navigation. Zoology. — “On the formation of red blood-corpuscles in the placenta of the flying maki (Galeopithecus). By Prof. A. A. W. Husrncur. (Communicated in the meeting of March 30, 1907). At the meeting of November 26, 1898, I made a communication on the formation of blood in the placenta of Tarsius and other mammals, which was later completed by a more extensive paper, containing many illustrations (Ueber die Entwicklung der Placenta von Tarsius und Tupaja, nebst Bemerkungen über deren Bedeutung als hämatopoietische Organe; Report 4'" Intern. Congress of Zoology, Cambridge 1898). The facts observed by me and the interpretation founded on them, have not until now been generally accepted, and in a recent very extensive discussion of the position of the problem concerning the origin of the red blood-corpuscles in the 14% volume of the “Ergebnisse der Anatomie und Entwicklungsgeschichte” (Wies- baden 1905), by F. Wemenreicu, the author, when mentioning my views, emits the supposition that | mixed up phagocytic and haemato- poietic processes. This conclusion was not based on a renewed and critical exami- nation of the material, studied by me. I have regretted this, since I have pointed out clearly and repeatedly that the numerous prepara- ( 874) tions at Utrecht concerning this and other embryological problems are always available for comparative and critical work, also for those who do not share my views. Moreover it appears from the literature, mentioned in WeIDENREICH’s paper, that the more extensive and illu- strated article, quoted above, has remained unknown to him. All this would not have induced me to return to this subject once more, were it not for the fact that during the last months I have become acquainted with the placentation-phenomena of a totally different mammal in which these phenomena have never yet been studied, namely Galeopithecus volans, which, like Tarsius, Nyeti- cebus, Tupaja and Manis, was collected by me in the Indian Archipelago in 1890—1891 as extensively as possible for embryological purposes. During the first origin of the placenta of this rare and in many respects primitive mammal‘), phenomena are observed which elucidate the process of blood-formation in the placenta in such aa uncommonly clear manner that in this case it will be difficult to deny the evidence. The formation of blood in the placenta of Galeopithecus may be said to take place according to a much simpler plan than in Tarsius, although the principal outlines remain the same and here also the non-nucleate haemoglobine-carrying blood-corpuscles must be regarded not as modified cells but as nuclear derivatives. Likewise the placenta of Galeopithecus bears testimony that not only the maternal mucosa but also the embryonic trophoblast takes part in the blood-formation, while the thus formed blood-corpuscles — also those that are furnished by embryonic tissue — circulate in the maternal blood-vessels only. In Galeopithecus the process is simpler especially in this respect that here no megalokaryocytes play a part in the formation of blood, so that it is less easy as Weripenreicn did — to regard blood- corpuscles that are set free (such as we notice it in Tarsius, when the big lobed nuclei of these megalokaryocytes disintegrate) as being on the contrary devoured in that moment by phagocytosis! *) The haematopoiesis is started in Galeopitheeus in the following manner. At about the same time that the young germinal vesicle, which has just gone through the two-layered gastrulation stage (gastru- lation by delamination *)), has attached itself to the surface of the strongly folded and swollen maternal mucous membrane, this mucous 1) W. Lecue is inclined (Ueber die Säugethiergattung Galeopithecus, Svenska Akad. Handl. Bd. 21, N°. 11, 1886) to see in Galeopithecus a form which must be piaced in the neighbourhood of the ancestral form of the bat. 2) Sectional series of Tarsius of a later date give a still clearer image than those which served for my figures of 1898. 3) See on this point Anatomischer Anzeiger Bd. 26, 353. ( 875 ) membrane reacts in the manner, well-known in other mammals (Tarsius, hedgehog, rabbit, bat, ete.) by perceptible changes in the uterine glands in the vicinity of this place of attachment and by the formation of so-called trophospongia-tissue, consisting of a modi- fication of the interglandular connective tissue, to which are added proliferations of uterine and glandular epithelium. As the final product of these preliminary phenomena we now see that a part of the maternal mucosa where the germinal vesicle has coalesced with the mucosa, presents a more compact proli- feration, while nearer the periphery the uterine glands, by strong dilatation of their lumen, differ clearly from the other uterine glands, as this is also the case in Tarsius, Lepus and other mam- mals during early pregnancy. The dilated glands may be followed up to their mouth; this mouth, however, no longer connects the glandular lumen with the uterine lumen, since in this place the embryonic trophoblast has disturbed the connection and covers the mouths of the glands. This trophoblast now also shows unmistakable signs of cell-prolifer- ation, although it does not at once attack and destroy the maternal epithelium, as in the hedgehog, Tarsius, Tupaja, ete. but rather finds itself facing this maternal epithelium in full proliferation, in the manner stated by me also for Sorex). Instead of being closely adjacent, however, spaces are left open from the beginning between trophoblast and trophospongia, which spaces are partly mutually connected and partly are subdivided into smaller compartments by trophoblastic villi, attaching themselves to the trophospongia-tissne. In this manner the free surface of the trophoblast, facing the embryo, obtains a knobbed appearance. *) Already in early developmental stages, when there is as yet no question of the folding off of the embryo and long before blood- carrying allantoic villi have become interlocked with these tropho- blastic villi for the further completion of the placenta, we find in the spaces between trophoblast and trophospongia numerous blood- corpuscles of which we can not say that they have been carried thither by maternal vessels exclusively, although there can be no doubt that a connection between these spaces and the maternal vascular system is established at an early date. In the manner, indicated above, these spaces communicate also with the uterine glands which are here dilated. And in these glands as well as in 1) Quarterly Journal of Microscopical Science, vol. 35. *) Certain modifications which L cbserved when the germinal vesicle develops in a uterus which is still in the puerperal stage, may be left out of account here, ( 876 ) the interglandular tissue and in the cells, lining the just mentioned spaces, phenomena take place which force us to the conclusion that a great number of these blood-corpuscles originate in loco. When we follow these phenomena up to their earliest appearance, we find that in the dilated glands in many places compact cell-heaps are formed, which sometimes lie quite loose in the gland, but in other cases are still found in direct connection with the cell-lining of the gland. We rust assume that this latter condition represents the original one and that consequently we have here an epithelial proli- feration by which new cell-material is carried into the region of the future placenta. The final product of these lumps of tissue, which in early stages appear so distinctly as cell-heaps, is an agglomerate of non-nucleated blood-corpuscles. The gradual transition of the nucleate cells into the blood-dises may be followed step by step by successively comparing preparations of the youngest and subsequent stages: often in one preparation all transitions are found together. It then appears that the conclusions I drew for Tarsius and Tupaja in 1898 are confirmed here, viz. that the blood-dises are produced by gradual transitions from the modified nuclei of the above-mentioned cell-heaps and that in this process transitional stages are generally found, comparable to what I called ‘‘haematogonia” in the above-quoted paper. They re- semble polynuclear leucocytes from which they may be distinguished, however (also according to Maxtmow and SIEGENBEEK VAN HEUKELOM; see report of the meeting of the Amsterdam Academy of Nov. 26, 1898), by certain characteristics. This phenomenon has been more fully investigated by Porsakorr, who also regards the non-nucleate corpuscles as nuclear derivatives and not as cells, deprived of their nuclei. In his paper*) numerous illustrations are given of stages corresponding to my haematogonia. It appears from the literature, mentioned by Porsakorr that my paper of 1898, preceding his publication, was unknown to him: the concordant results which we have obtained at an earlier date, are confirmed in a striking manner by the phenomena seen in Galeopithecus. But blood-corpuseles are also produced by other sources besides these epithelial glandular proliferations. Between the dilated glands we find in Galeopitheeus in the trophospongia-tissue very conspicuous groups of large cells with a big, but circular nucleus. They show a tendency to lie together in nests, which nests are more or less kept together by elongated cells, forming a spurious wall which distantly remind us of an endothelium. 1) Biologie der Zelle. In Arch. f. Anat. u. Phys. Abth. 1901. Pl. I and II. ( 877 ) These cells also are gradually dissolved into blood-corpuscles : as the uterus grows and the trophospongia passes through its successive developmental stages, they disappear: the blood-corpuscles which owe their existence to them, fall into the above-mentioned spaces, from whence they are taken up in the further circulation. The intermediate stages that can be observed in this way of blood-formation, are in fact an increase of nuclei by amitosis, as was also described by PoLJAKOFF and later a gradual formation from these nuclear derivates of non-nucleated blood-dises. To these two processes of blood-formation in the placenta of Galeopitheeus a third must be added in which not the mother is the active agent, as in the two former cases, but the embryonic trophoblast. Of this trophoblast we described above how it forms the bottom of the cavities into which the newly-formed blood-corpus- cles are discharged, and how it coalesces with the maternal trophos- pongia to such an extent that for many cells, which here are closely adjacent, it is impossible to determine whether they take their origin in the mother or in the trophoblast of the germinal vesicle. Yet in regard to the wall of the cavities, which separates them from the lumen of the uterus, there can be no doubt that we have here trophoblastic tissue only. About the active proliferation of this trophoblast tissue there is no doubt, no more than about the question whether the numerous parts of this trophoblast that project into the cavities, partake in the haematopoiesis. As soon as these parts are examined with strong powers it is quite evident that here the nuclei of the trophoblast cells undergo similar modifications as were described above and that the final product of these modifications are again red non-nucleated blood-corpuscles which are added to those already present and originating from the mother. Now these corpuscles are, in the same way as I observed ten years ago in Tarsius and Tupaja, set free into the maternal circulation and carried along by it. On the theoretical significance of the fact that the germinal vesicle takes an active and important part in increasing the number of units for the transport of oxygen in the maternal blood, I will not expatiate here. And for the histological details of the formation of the bloodplates, resp. non-nucleated blood-corpuscles from an originally normal cell- nucleus, I refer to the coloured figures of pl. I and II of Porsakorr’s paper in the 1901 volume of the Arch. f. Anat. u. Phys. (Anat. Abth.). With his illustrations I can identify everything I have ob- served in Galeopithecus. While in a very few cases there seems to be a possibility that the blood-corpuscle owes its existence to a ( 878 ) change of the nucleus im its entirety, in the vast majority of cases a distinct amitotic disintegration is observed, the number of fragments varying, but generally lying between three and five. As the already modified nucleus dissolves into these fragments the comparability with polynuclear leucocytes seems more obvious, and the colour as a rule approaches more and more to that which the blood-corpuseles themselves assume in the artificially fixed preparation. The same fact was stated by me also for Tarsius in 1898 and figured on Pl. 14 figs. 91—96. Finally I point out, since my results and those of PorJakorr agree in so many respects, that also Rerrerer in the volume for 1901 of the Journal de Anatomie et de la Physiologie (Structure, développement et fonction des ganglions lymphatiques, p. 700) has obtained similar results and is inclined to assume a still closer genetic relationship between polynuclear leucocytes and haematogonia when he declares that the leucocytes, liberated from lymphatic glands “finissent par se convertir, dans la lymphe ou le sang, en hématies grace a la transformation hémoglobique de leur noyau Thus my observations on Galeopitheous form a link in the chain, which begins with Heinrich Mürrer in 1845 (Zeitschrift für rationelle Medicin vol. 3. p. 260) was then continued and upheld by W HARTON Jones (Phil. Trans. 1846, p. 65 and 71) and Huxrey (Lessons in Elementary Physiology, 1866, p. 63) and which, since in 1898 Tarsius added another link, has with increasing weight bound up the question of the origin of the non-nucleated blood-corpnseles in mammals to the conception that these elements in the mammalian body are not equivalent with cells, but must be regarded as nuclear derivatives. (May 24, 1907). CONTENTS. ABSORPTION BANDS (Wave-lengths of formerly observed emission and) in the infra- red spectrum. 706. acip (On the nitration of phthalic acid and isophthalic), 286. ACIDs (The six isomeric dinitrobenzoic). 280. AIR (The preparation of liquid) by means of the cascade process. 177. AMBOCEPTORS (On the) of an anti-streptococcus serum. 336. : AMMONIA aud Amines (On the action of bases,) on s.trinitrophenyl-methylnitra- mine. 704. AMYRIN (On z- and @-) from bresk. 471. Anatomy. A. J. P. van DEN BROEK: “On the relation of the genital ducts to the genital gland in marsupials”. 396. — B. van Tricut: “On the influence of the fins upon the form of the trunk- myotome”’. 814. — L. J. J. Muskens: “Anatomical research about cerebellar connections” (3rd Com- munication). 819. ANILINEHYDROCHLORIDE (Three-phaselines in chloralaleoholate and). 99. ANTHERS (On the influence of the nectaries and other sugar-containing tissues in the flower on the opening of the). 390. ANTHRACOsIS (On the origin of pulmonary). 673. ANTIMONITE from Japan (On the influence which irradiation exerts on the electrical conductivity of). $09. ARRHENIUS (SVANTE) and H. J. HAMBURGER. On the nature of precipitin- reaction. 33. Astronomy. J. Stein: “Observations of the total solar eclipse of August 30, 1905 at Tortosa (Spain).”” 45. — A, PANNEKOEK: “The luminosity of stars of different types of spectrum,” 134, — A, PANNEKOEK: “The relation between the spectra and the colours of the stars.” 292. — J. A. C. OupeMans: “Mutual occultations and eclipses of the satellites of Jupiter in 1908.” 304, 2nd part. 444. — H. J. Zwiers: “Researches on the orbit of the periodic comet Holmes and on the perturbations of its elliptic motion.” LV. 414. 60 Proceedings Royal Acad. Amsterdam. Vol. IX. lr CONTENTS. Astronomy. H. G. VAN DE SANDE BAKHUYZEN: “On the astronomical refractions correspond- ing to a distribution of the temperature in the atmosphere derived from balloon ascents.” 578. ATMOSPHERE (On the astronomical refractions corresponding to a distribution of the temperature in the) derived from balloon ascents. 578. BAKHUIS ROOZEBOOM (H. W.) presents a paper of Dr. A. Smits: “On the introduction of the conception of the solubility of metal ions with electromotive equilibrium.” 2. — presents a paper of J. J. van Laar: “On the osmotic pressure of solutions of non-electrolytes, in connection with the deviations from the laws of ideal gases.” 53. — Three-phaselines in chloralalcoholate and anilinehydrochloride. 99, — presents a paper of Dr. F. M, Jararr: “On a substance which possesses nume- rous different liquid. phases of which three at least are stable in regard to the isotropous liquid.” 359. — The behaviour of the halogens towards each other. 363. BAKHUYZEN (H. G. VAN DE SANDE). v. SANDE BAKHUYZEN (H. G. VAN DE). BALLOON ASCENTS (On the astronomical refractions corresponding to a distribution of the temperature in the atmosphere derived from). 578. BATAVIA (On magnetic disturbances as recorded at). 266. satus (How to obtain) of constant and uniform temperature by means of liquid hydrogen. 156. BEMMELEN (W. VAN). On magnetic disturbances as recorded at Batavia. 266. BENZENE-DERIVATIVES (On a new case of form-analogy and. miscibility of position- isomeric), and on the crystal forms of the six nitrodibromobenzenes. 26. BINARY MIXTURE (On the shape of the three-phase line: solid-liquid-vapour for a). 639. — (The shape of the empiric isotherm for the condensation of a). 750. BINARY MIXTURES (The shape of the spinodal and plaitpoint curves for) of normal sub- stances. 4th Communication. The longitudinal plait. 226. — (On the gas phase sinking in the liquid phase for). 501. — (On the conditions for the sinking and again rising of the gas phase in the liquid phase for). 508. Continued. 660. — (A remark on the theory of the \p-surface for). 524. — (Contribution to the theory of). 621. IL, 727. ILL. 826. — (Isotherms of diatomic gases and their). VL. Isotherms of hydrogen between — 104° ©. and — 217° C. 754. BLANKSMA (J. J.). Nitration of meta-substituted phenols. 278. BLOOD-CORPUSCLES (On the formation of red) in the placenta of the flying maki (Galeopithecus). 873. BOEKE (J.). Gastrulation and the covering of the yolk in the teleostean egg. 800. BOËSEKEN (J.). On catalytic reactions connected with the transformation of yellow phosphorus into the red modification. 613. BOLK (L.) presents a paper of Dr. A. J. P. van DEN Broek: “On the relation of the genital ducts to the genital gland in marsupials.” 396. CO NT EN, TS. iI Botany. M. Nizuwenuuis- von UEXKÜLL-GÜLDENBAND : “On the harmful consequences of the secretion of sugar with some myrmecophilous plants”. 150. — W. Burcx: “On the influence of the nectaries and other sugar-containine 5 le) tissues in the flower on the opening of the anthers’. 390. BRAAK (c.) and H. KAMERLINGH Onnes. On the measurement of very low tem- peratures. XILL. Determinations with the hydrogen thermometer. 367. XIV. Reduction of the readings of the hydrogen thermometer of constant volume to the absolute scale. 775. — Isotherms of diatomic gases and their binary mixtures. VI. Isotherms of hy- drogen between — 104°C and — 217°C. 754. BRESK (On z-and j-amyrin from). 471, BROEK (A. J. P, VAN DEN). On the relation of the genital ducts to the genital gland in marsupials. 396, BROUWER(L. E. J.), Polydimensional vectordistributions. 66. — The force field of the non-Kuclidean spaces with negative curvature. 116. — The force field of the non-Kuclidean spaces with positive curvature. 250. BURCK (w.). On the influence of the nectaries and other sugar-containing tissues in the flower on the opening of the anthers. 390. BUTYRIC ESTER of dihydrocholesterol, (On the anisotropous liquid phases of the) and on the question as to the necessary presence of an ethylene double bond for the occurrence of these phenomena. 701. CARDIAC ACTION (An investigation on the quantitative relation between vagus stimu- lation and). 590. CASCADE PROCESS (The preparation of liquid air by means of the). 177. cases (A few remarks concerning the method of the true and false). 222. CATALYTIC REACTIONS (On) connected with the transformation of yellow phosphorus into the red modification. 613. CEREBELLAR CONNECTIONS (Anatomical research about). 3rd Communication. 819. Chemistry. A. Smits: “On the introduction of the conception of the solubility of metal ions with electromotive equilibrium.” 2, — J. Moun van Cuarante: “The formation of salicylic acid from sodium pheno- late.” 20. — F. M. JarGer: “On the crystal-forms of the 2.4 dinitroaniline-derivatives, sub- stituted in the NH,-group.” 23. — J. J. van Laar: “On the osmotic pressure of solutions of non-electrolytes, in connection with the deviations from the laws of ideal gases.” 53. — H. W. Bakuuts Roozesoom: “Three-phaselines in chloralaleoholate and aniline- hydrochloride.” 99. — P. van RompBurcu: “Triformin (Glyceryltriformate).”’ 109. — P. van RomBureH and W. van Dorssen: “On some derivatives of 1-3-5- hexatriene.” 111. — J. J. van Laar: “The shape of the spinodal and plaitpoint curves for binary mixtures of normal substances. 4th Communication. The longitudinal plait” 226. — J. J. Buanksma: “Nitration of meta-substituted phenols,” 278. 60* bf CON TENTS. Chemistry. A. F. HoLLeMAN and H. A, Sirxs: “The six isomeric dinitrobenzoic acids.” 280. — A. F. HOLLEMAN and J. Hursinea: “On the nitration of phthalic acid and isophthalic acid’? 286. — R. A. Weerman; “Action of potassium hypochlorite on cinnamide.” 303. — F. M. Jarcer: “On a substance which possesses numerous different liquid phases of which three at least are stable in regard to the isotropous liquid.” 359. — H. W. Bakuuis Roozesoom: “The behaviour of the halogens towards each other,” 363. — N. H. Conen: “On Lupeol.” 466. — N. H. COHEN: “On z- and B-amyrin from bresk.”’ 471. — Ff. M. JarGEr: “On substances which possess more than one stable liquid stat , and on the phenomena observed in anisotropous liquids.” 472. — F. M. JAEGER: “On irreversible phase-transitions in substances which may exhibit more than one liquid condition.” 483. — A. F. HorLEMAN and G. L. VOERMAN: “z- and @B-thiophenic acid.” 514. — A. P. N. FRANCHIMONT: “Contribution to the know